.*>: 


This  book  is  DUE  on  the  last  date  stamped  below 


2  0  1025 


MAR 


81920 


ocr  s 


APR  i  4  1932 


»AR      8  1943 


ELEMENTARY    TREATISE 


ON 


NATURAL  PHILOSOPHY. 


BY 

A.  PRIVAT   DESCHANEL, 

FORMERLY    PROFESSOR   OF   PHTSICS   IN   THE   LTCEE   LOUIS-LE -GRAND, 
INSPECTOR  OF   THE   ACADEMY  OF   PARIS. 


TRANSLATED    AND    EDITED,   WITH    EXTENSIVE    MODIFICATIONS, 

BY  J.  D.  EVERETT,  M.  A.,  D.  C.  L.,  F.  R.  S.,  F.  R.  S.  E., 

PROFESSOR    OF    NATURAL    PHILOSOPHY    IX    THE    QUEEN'S    COLLEGE,  BELFAST. 


IN     FOUR     PARTS. 

PART    I. 

MECHANICS,    HYDROSTATICS,    AND    PNEUMATICS, 

ILLUSTRATED    BY 
180    ENGRAVINGS    ON    WOOD,    AND    ONE    COLORED    PLATE. 

REVISED    EDITION. 


NEW  YORK : 
D.     APPLETON    AND    COMPANY, 

1,  3,    AND    5   BOND    STREET. 

1884. 


56207 


AUTHOR'S  PEEFACE. 


C^THE  importance  of  the  study  of  Physics  is  now  generally  acknowledged.     Besides 

(f)  the  interest  of  curiosity  which  attaches  to  the  observation  of  nature,  the  experi- 

rf>   mental  method   furnishes  one  of  the  most  salutary  exercises   for  the  mind — 

~~   constituting  in  this  respect  a  fitting  supplement  to  the  study  of  the  mathematical 

sciences.     The  method  of  deduction  employed  in  these  latter,  while  eminently 

adapted  to  form  the  habit  of  strict  reasoning,  scarcely  affords  any  exercise  for 

the  critical   faculty  which  plays  so  important  a  part  in  the  physical  sciences. 

In  Physics  we  are  called  upon,  not  to  deduce  rigorous  consequences  from  an 

ri      absolute  principle,  but  to  ascend  from  the  particular  consequences  which  alone 

r^      are  known  to  the  general  principle  from  which  they  flow.     In  this  operation 

^      there  is  no  absolutely  certain  method  of  procedure,  and  even  relative  certainty 

can  only  be  attained  by  a  discussion  which  calls  into  profitable  exercise  all  the 

faculties  of  the  mind. 

Be  this  as  it  may,  physical  science  has  now  taken  an  important  place  in  educa- 

$»      tion,  and  plays  a  prominent  part  in  the  examinations  for  the  different  university 

degrees.     The   present   treatise   is   intended   for   the   assistance  of  young   men 

^      preparing  for  these  degrees;   but  I  trust  that  it  may  also  be  read  with  profit 

^5>    by  those  persons  who,  merely  for  purposes  of  self-instruction,  wish  to  acquire 

$      accurate  knowledge  of  natxiral   phenomena.     Having   for  nearly  twenty  years 

been  charged  with  the  duty  of  teaching  from  the  chair  of  Physics  in  one  of  the 

lyceums  of  Paris,  I  have  been  under  the  necessity  of  making  continual  efforts 

to  overcome  the  inherent  difficulties  of   this  branch  of  study.     I   have  endea- 

•}      voured  to  turn  to  account  the  experience  thus  acquired  in  the  preparation  of  this 

j      volume,  and  I  shall  be  happy  if  I  can  thus  contribute  to  advance  the  taste  for 

a  science  which  is  at  once  useful  aud  interesting. 

L*  I  have  made  very  limited  use  of  algebra.     Though  calculation  is  a  precious 

and  often  indispensable  auxiliary  of  physical  science,  the  extent  to  which  it 
can  be  advantageously  employed  varies  greatly  according  to  circumstances.  There 
are  in  fact  some  phenomena  which  cannot  be  really  understood  without  having 
recourse  to  measurement:  but  in  a  multitude  of  cases  the  explanation  of 
phenomena  can  be  rendered  evident  without  resorting  to  numerical  expression. 

The  physical  sciences  have  of  late  years  received  very  extensive  developments. 
Facts  have  been  multiplied  indefinitely,  and  even  theories  have  undergone  great 
modifications.  Hence  arises  considerable  difficulty  in  selecting  the  most  essential 
points  and  those  which  best  represent  the  present  state  of  science.  I  have  done 
my  best  to  cope  with  this  difficulty,  and  I  trust  that  the  reader  who  attentively 
peruses  my  work,  will  be  able  to  form  a  pretty  accurate  idea  of  the  present 
position  of  physical  science. 


TKANSLATOK'S    PREFACE 

TO   THE   SIXTH   EDITION. 


I  DID  not  consent  to  undertake  the  labour  of  translating  and  editing  the 
"TRAITS  ELEMENTAIRE  DE  PHYSIQUE"  of  Professor  Deschanel  until  a 
careful  examination  had  convinced  me  that  it  was  better  adapted  to  the 
requirements  of  my  own  class  of  Experimental  Physics  than  any  other 
work  with  which  I  was  acquainted;  and  in  executing  the  translation  I 
steadily  kept  this  use  in  view,  believing  that  I  was  thus  adopting  the  surest 
means  of  meeting  the  wants  of  teachers  generally. 

In  the  original  English  edition,  the  earlier  portions  consisted  of  a  pretty 
close  translation  from  the  French;  but  as  the  work  progressed  I  found  the 
advantage  of  introducing  more  considerable  modifications;  and  Parts  III. 
and  IV.  were  to  a  great  extent  rewritten  rather  than  translated.  I  have 
now,  in  like  manner,  rewritten  Part  I.,  and  trust  that  in  its  amended  form 
it  will  be  found  better  adapted  than  before  to  the  wants  of  English  teachers. 
Several  additional  subjects  have  been  introduced,  and  the  order  of  tho 
chapters  has  been  rearranged. 

The  marks  of  distinction  which  were  made  in  the  earlier  editions 
between  new  and  old  sections  have  now  been  dropped;  but  Professor 
Deschanel's  foot-notes  are  still  distinguished  by  the  initial  "D."  The 
numbering  of  the  sections  is  entirely  new. 

All  accurate  statements  of  quantities  have  been  given  in  the  C.G.S. 
(Centimetre-Gramme-Second)  system,  which,  by  reason  of  its  simplicity  and 
of  the  sanction  which  it  has  received  from  the  British  Association,  and  the 
Physical  Society  of  London,  is  coming  every  day  into  more  general  use,  but 
rough  statements  of  quantity  have  generally  been  expressed  in  British 
units  as  being  more  familiar.  A  complete  table  for  the  conversion  of 
French  and  English  measures  will  be  found  at  the  end  of  the  Table  of 
Contents. 

In  Part  II,  the  subject  of  Heat  as  a  measurable  Quantity  is  introduced 
at  a  much  earlier  stage  than  before,  the  chapter  on  Calorimetry  being 
placed  immediately  after  those  on  Thermometry  and  Expansion.  Latent 
Heat  and  Heat  of  Combination  are  not  now  included  in  this  chapter,  but 
are  treated  later  in  connection  with  the  subjects  of  Fusion,  Vaporization, 
and  Thermo-dynamics. 


TRANSLATOR  S   PREFACE.  V 

Among  the  new  matter  may  be  mentioned: — 

An  investigation  of  the  temperature  of  minimum  apparent  volume  of 
water  in  a  glass  envelope  ; 

An  account  of  Guthrie's  results  on  the  freezing  of  brine ; 

A  proof  that  the  pressure  of  vapour  in  the  air  at  any  time  is  equal  to  the 
maximum  pressure  for  the  dew-point; 

Descriptions  of  Dines'  hygrometer,  and  of  Symons'  Snowdon  rain-gauge  ; 

A  full  explanation  of  " Diff usivity "  or  " Thermometric  Conductivity;" 

Some  recent  results  on  the  conductivity  of  rocks,  and  on  the  conductivity 
of  water ; 

A  note  on  the  mathematical  discussion  of  periodical  variations  of  under- 
ground temperature; 

A  proof  of  the  formula  for  the  efficiency  of  a  perfect  thermo-dynamic 
engine ; 

Several  investigations  relating  to  the  two  specific  heats  of  a  gas,  and  to 
adiabatic  changes  in  gases,  liquids,  and  solids ; 

A  description  of  the  modern  Gas  Engine. 

Every  chapter  has  been  carefully  revised,  with  a  view  to  clearness, 
accuracy,  and  consolidation;  and  the  result  has  been  that,  with  the  excep- 
tion of  Melloni's  experiments,  and  the  Steam  Engine,  the  treatment  of 
nearly  every  subject  has  been  materially  changed. 

Part  III.  also  contains  extensive  changes. 

In  the  electro-statics,  the  chapter  on  potential  has  been  recast  and  made 
more  demonstrative.  There  are  also  additions  relating  to  Dr.  Kerr's  dis- 
coveries, charge  by  cascade,  and  some  minor  points. 

Under  the  head  of  Magnetism,  investigations  have  been  introduced 
relating  to  bih'lar  suspension,  and  to  the  directive  tendency  of  soft-iron 
needles. 

In  the  department  of  Current  Electricity,  there  has  been  a  complete 
rearrangement  of  subjects.  The  chemical  relations  of  the  current  are 
discussed  as  early  as  possible,  while  thermo-electricity  is  reserved  for  a 
chapter  on  relations  between  electricity  and  heat.  The  chapter  on  induced 
currents,  which  was  formerly  the  last  of  all,  has  been  put  next  to  that  on 
electro-dynamics,  and  is  followed  by  two  chapters  on  telegraphs  and  other 
applications  of  electricity.  Additional  matter  has  been  introduced  under 
the  following  heads : — 

General  law  for  magnetic  force  due  to  current  in  given  circuit; 

Helmholtz's  galvanometer ; 

Swing  produced  by  instantaneous  current ; 

The  galvanometer  a  true  measurer  of  current ; 

Rowland's  experiment  on  the  motion  of  a  charged  body ; 

Flante's  secondary  battery ; 

Chemical  relations  of  electro-motive  force ; 

Resistance  coils  and  boxes ; 

Wheatstone's  bridge,  and  conjugate  branches ; 


vi  TRANSLATOR'S  PREFACE. 

Clark's  method  for  electro-motive  force ; 

Thomson's  method  for  resistance  of  galvanometer; 

Mance's  method  for  resistance  of  battery  ; 

Thermo-electric  diagrams ; 

Convection  of  heat  by  electricity ; 

Pyro-electricity ; 

Effect  of  light  on  resistance  of  selenium ; 

Deduction  of  law  of  induced  currents  from  electro-dynamic  law; 

Superposition  of  tubes  of  force ; 

Stratified  discharge  from  galvanic  battery ; 

Siemens'  and  Gramme's  magneto-electric  machines ; 

Cowper's  writing  telegraph ; 

Duplex  telegraphy ; 

Edison's  electric  pen ; 

The  telephone,  the  microphone,  and  the  induction  balance. 

A  collection  of  examples  on  electricity  has  been  added. 

Part  IV.  contains  no  radical  changes.  The  numbering  of  the  chapters 
and  sections  has  been  altered  to  make  it  consecutive  with  the  other  three 
Parts,  but  there  has  been  no  rearrangement. 

Additions  have  been  made  under  the  following  heads  (those  marked  with 
an  asterisk  Avere  introduced  in  a  previous  edition) : — 

Mathematical  note  on  stationary  undulation ; 

Edison's  phonograph ; 

Michelson's  measurement  of  the  velocity  of  light; 

Astronomical  refraction ; 
*Eefraction  at  a  spherical  surface; 

Refraction  through  a  sphere; 

Brightness  of  image  on  screen ; 

Field  of  view  in  telescope ; 
*Curved  rays  of  sound ; 
*Retardatiou-gratings  and  reflection-gratings ; 

Kerr's  magneto-optic  discoveries; 

besides  briefer  additions  and  emendations  which  it  would  be  tedious  to 
enumerate. 

The  whole  volume  has  been  minutely  revised;  and  a  copious  collection 
of  examples  arranged  in  order,  with  answers,  has  been  introduced  at  the 
end  of  each  Part,  in  place  of  the  "  Problems  "  (translated  from  the  French) 
which  appeared  in  some  of  the  earlier  editions. 

The  dates  of  revision  of  the  four  Parts  were,  October,  1879,  November 
1880,  Docember,  1880,  and  May,  1881. 

J.  D  E 

BELFAST,  September,  1SS1. 


CONTENTS-PAKT   I. 


(THE  NUMBERS  REFER  TO  THE  SECTIONS.) 


CHAPTER  I.    INTRODUCTORY. 

Natural  History  and  Natural  Philosophy,  1,  2.     Divisions  of  Natural  Philosophy,  3. 

CHAPTER  II.    FIRST  PRINCIPLES  OF  DYNAMICS,  STATICS. 

Force,  4.  Translation  and  rotation,  5,  6.  Instruments  for  measuring  force,  7.  Gravita- 
tion units  of  force,  8.  Equilibrium ;  Statics  and  kinetics,  9.  Action  and  reaction, 
10.  Specification  of  a  force,  point  of  application,  line  of  action,  11.  Rigid  body,  12. 
Equilibrium  of  two  forces,  13.  Three  forces  in  equilibrium  at  a  point,  14.  Resul- 
tant and  components,  15.  Parallelogram  of  forces,  16.  Gravesande's  apparatus,  17. 
Resultant  of  any  number  of  forces  at  a  point,  18.  Equilibrium  of  three  parallel 
forces,  19.  Resultant  of  two  parallel  forces,  20.  Centre  of  two  parallel  forces,  21. 
Moments  of  resultant  and  components  equal,  22.  Resultant  of  any  number  of 
parallel  forces  in  one  plane,  23.  Moment  of  a  force  about  a  point,  24.  Arithmetical 
lever,  25.  Couple,  26.  Composition  of  couples ;  Axis  of  couple,  27.  Resultant  of 
force  and  couple  in  same  plane,  28.  General  resultant  of  any  number  of  forces ; 
Wrench,  29.  Application  to  action  and  reaction,  30.  Resolution,  31.  Rectangular 
resolution ;  Component  of  a  force  along  a  given  line,  32. 

CHAPTER  III.     GRAVITY. 

Direction  of  gravity ;  Neighbouring  verticals  nearly  parallel,  33.  Centre  of  gravity,  34. 
Centres  of  gravity  of  volumes,  areas,  and  lines,  35.  Methods  of  finding  centres  of 
gravity,  36.  Centre  of  gravity  of  triangle,  37.  Of  pyramids  and  cones,  38,  39. 
Condition  of  standing  or  falling,  40.  Body  supported  at  one  point,  41.  Stability 
and  instability,  42.  Experimental  determination  of  centre  of  gravity,  43,  44.  Work 
done  against  gravity,  45.  Centre  of  gravity  tends  to  descend,  46.  Work  done  by 
gravity,  47.  Work  done  by  any  force,  48.  Principle  of  work;  Perpetual  motions, 
49.  Criterion  of  stability,  50.  Illustration,  51.  Stability  where  forces  vary  abruptly, 
52.  Illustrations  from  toys,  53.  Limits  of  stability,  54. 

CHAPTER  IV.    THE  MECHANICAL  POWERS. 

Enumeration,  55.  Lever,  56-58.  Mechanical  advantage,  59.  Wheel  and  axle,  60. 
Pulleys,  61-63.  Inclined  plane,  64-66.  Wedge  and  screw,  67-69. 

CHAPTER  V.    THE  BALANCE. 

General  description,  70.  Qualities  requisite,  71.  Double  weighing,  72.  Investigation 
of  sensibility,  73.  Advantage  of  weighing  with  constant  load,  74.  Details  of  con- 
struction, 75.  Steelyard,  76. 

CHAPTER  VI.    FIRST  PRINCIPLES  OF  KINETICS. 

Principle  of  Inertia,  77.  Second  law  of  motion,  78.  Mass  and  momentum,  79.  Proper 
selection  of  unit  of  force,  80.  Relation  between  mass  and  weight,  81.  Third  law  of 


viii  TABLE   OF   CONTENTS. 

motion;  Action  and  reaction,  82.  Motion  of  centre  of  gravity  unaffected,  83. 
Velocity  of  centre  of  gravity,  84.  Centre  of  mass,  85.  Units  of  measurement,  86. 
The  C.G.S.  system;  the  dyne,  the  erg,  87. 

CHAPTER  VII.    LAWS  OF  FALLING  BODIES. 

Fall  in  air  and  in  vacuo,  88.  Mass  and  gravitation  proportional,  89.  Uniform  accelera- 
tion, 89.  Weight  of  a  gramme  in  dynes ;  Value  of  g,  91.  Distance  fallen  in  a  given 
time,  92.  Work  spent  in  producing  motion,  93.  Body  thrown  upwards,  94. 
Resistance  of  the  air,  95.  Projectiles,  96.  Time  of  flight,  and  range,  97.  Morin's 
apparatus,  98.  Atwood's  machine,  99.  Theory  of  Atwood's  machine,  100.  Uniform 
motion  in  a  circle,  101.  Deflecting  force,  102.  Illustrations,  stone  in  sling,  103. 
Centrifugal  force  at  the  equator,  104.  Direction  of  apparent  gravity,  105. 

CHAPTER  VIII.     THE  PENDULUM. 

Pendulum,  106.  Simple  pendulum,  107.  Law  of  acceleration  for  small  vibrations,  108. 
General  law  for  period,  109.  Application  to  pendulum,  110.  Simple  harmonic 
motion,  111.  Experimental  investigation  of  motion  of  pendulum,  112.  Cycloidal 
pendulum,  113.  Moment  of  inertia  about  an  axis,  114.  About  parallel  axes,  115. 
Application  to  compound  pendulum,  116.  Convertibility  of  centres,  117.  Centre  of 
suspension  for  minimum  period,  118.  Kater's  pendulum,  119.  Determination  of  g, 
120. 

CHAPTER  IX.    ENERGY. 

Kinetic  energy,  121.  Static  or  potential  energy,  122.  Conservation  of  mechanical  energy, 
123.  Illustration  from  pile-driving,  124.  Hindrances  to  availability  of  energy; 
Principle  of  the  conservation  of  energy,  125. 

CHAPTER  X.    ELASTICITY. 

Elasticity  and  its  limits,  126.  Isochronism  of  small  vibrations,  127.  Stress,  strain, 
coefficients  of  elasticity ;  Young's  modulus,  128.  Volume-elasticity,  129.  OZrsted's 
piezometer,  130. 

CHAPTER  XI.     FRICTION. 

Friction,  kinetical  and  statical,  131.  Statical  friction,  limiting  angle,  132.  Coefficient  = 
tan  0;  Inclined  plane,  133. 

CHAPTER  XII.     HYDROSTATICS. 

Hydrodynamics,  134.  No  statical  friction  in  fluids,  135.  Intensity  of  pressure,  136. 
Pressure  the  same  in  all  directions,  137.  The  same  at  the  same  level,  138.  Differ- 
ence of  pressure  at  different  levels,  139.  Free  surface,  140.  Transmissibility  of 
pressure ;  Pascal,  141.  Hydraulic  press,  142.  "  Principle  of  work  "  applicable,  143. 
Experiment  on  upward  pressure,  144.  Liquids  in  superposition,  145.  Two  liquids 
in  bent  tube,  146.  Pascal's  vases,  147.  Resultant  pressure  on  vessel,  148.  Back 
pressure  on  discharging  vessel,  149.  Total  and  resultant  pressures;  Centre  of 
pressure,  150.  Construction  for  centre  of  pressure,  151.  Whirling  vessel-  D'Alem- 
bert's  principle,  152. 

CHAPTER  XIII.     PRINCIPLE   OF  ARCHIMEDES. 

Resultant  pressure  on  immersed  bodies,  153.  Experimental  demonstration,  154.  Three 
cases  distinguished,  155.  Centre  of  buoyancy,  153,  155.  Cartesian  diver,  156 
Stability  of  floating  body,  157,  158.  Floating  of  needles  on  water  159 


TABLE   OF   CONTENTS.  IX 

CHAPTER  XIV.     DENSITY  AND   ITS   DETERMINATION. 

Absolute  and  relative  density,  160.  Ambiguity  of  the  word  "weight,"  161.  Determination 
of  density  from  observation  of  weight  and  volume,  162.  Specific  gravity  flask  for 
solids,  163.  Method  by  weighing  in  water,  164.  With  sinker,  165.  Densities  of 
liquids  measured  by  loss  of  weight  in  them,  166.  Measurement  of  volumes  of  solids 
by  loss  of  weight,  167.  Hydrometers,  168.  Nicholson's,  169.  Fahrenheit's,  170. 
Hydrometers  of  variable  immersion,  171.  General  theory,  172.  Beaume"'s  hydro- 
meters, 173.  Twaddell's,  174.  Gay-Lussac's  alcoholimeter,  175.  Computation  of 
densities  of  mixtures,  176.  Graphical  method  of  interpolation,  177. 

CHAPTER  XV.     VESSELS  IN   COMMUNICATION.     LEVELS. 

Liquids  tend  to  find  their  own  level;  Water-supply  of  towns,  178.  Water-level;  Levelling 
between  distant  stations,  179.  Spirit-level  and  its  uses,  180,  181. 

CHAPTER  XVI.     CAPILLARITY. 

General  phenomena  of  capillary  elevation  and  depression,  182.  Influencing  circum- 
stances, 183.  Law  of  diameters,  184.  Fundamental  laws  of  capillary  phenomena; 
Angle  of  contact;  Surface  tension,  185.  Application  to  elevation  and  depression  in 
tubes,  186.  Formula  for  normal  pressure  of  film,  187.  Film  with  air  on  both  sides, 
188.  Drops,  189.  Pressure  in  a  liquid  whose  surface  is  convex  or  concave,  190. 
Interior  pressure  due  to  surface  action  when  surface  is  plane,  191.  Phenomena  illus- 
trative of  differential  surface  tensions;  Table  of  tensions,  192.  -Endosmose  and 
diffusion,  193. 

CHAPTER  XVII.     THE  BAROMETER. 

Expansibility  of  gases,  194.  Direct  weighing  of  air,  195.  Atmospheric  pressure,  196. 
Torricellian  experiment,  197.  Pressure  of  one  atmosphere,  198.  Pascal's  experi- 
ment on  Puy  de  Dome,  199.  Barometer,  200.  Cathetometer,  201.  Fortin's 
Barometer ;  Vacuum  tested  by  metallic  clink,  202.  Float  adjustment,  203.  Baro- 
metric corrections;  Temperature;  Capillarity;  Capacity;  Index  errors;  Reduction 
to  sea-level;  Intensity  of  gravity;  and  reduction  to  absolute  measure,  204.  Siphon, 
wheel,  and  marine  barometers,  205.  Aneroid,  206.  Counterpoised  barometer; 
King's  barograph;  Fahrenheit's  multiple-tube  barometer,  207.  Photographic  regis- 
tration, 208. 

CHAPTER  XVHI.    VARIATIONS  OF  THE  BAROMETER, 

Measurement  of  heights  by  the  barometer,  209.  Imaginary  homogeneous  atmosphere, 
210.  Geometric  law  of  decrease,  211.  Computation  of  pressure-height,  212.  For- 
mula for  determining  heights  by  the  barometer,  213.  Diurnal  oscillation,  214. 
Irregular  variations,  215.  Weather  charts,  216. 

CHAPTKR  XIX.    BOYLE'S  (OR  MARIOTTE'S)   LAW. 

Boyle's  law,  217.  Boyle's  tube,  218.  Unequal  compressibility  of  different  gases,  219, 
220.  Regnault's  experiments,  221.  Results,  222.  Manometers  or  pressure  gauges, 
223.  Multiple-branch  manometer,  224.  Compressed  air  manometer,  225.  Metallic 
manometers,  226.  Pressure  of  gaseous  mixtures,  227.  Absorption  of  gases  by 
liquids  and  solids,  228. 

CHAPTER  XX.    AIR-PUMP. 

Air-pump,  229.  Theoretical  rate  of  exhaustion,  230.  Mercurial  gauges,  231.  Admission 
cock,  232.  Double-barrelled  pump,  233.  Single  barrel  with  double  action,  234.  English 


X  TABLE   OF   CONTEXTS. 

forms,  235.  Experiments;  Burst  bladder  ;  Magdeburg  hemispheres  ;  Fountain,  236. 
Limit  to  action  of  pump  and  its  causes,  237.  Kravogl's  pump,  238.  Geissler's,  239, 
Sprengel's,  240.  Double  exhaustion,  241.  Free  piston,  242.  Compressing  pump, 
2-13.  Calculation  of  its  effect,  244.  Various  contrivances  for  compressing  air,  245. 
Practical  applications  of  air-pump  and  compressing  pump,  246. 

CHAPTER  XXI.    UPWARD  PRESSURE  OF  THE  AIR. 

Baroscope,  247.  Principle  of  balloons,  248.  Details,  249.  Height  attainable  by  a  given 
balloon,  250.  Effect  of  air  on  apparent  weights,  251. 

CHAPTER  XXII.    PUMPS  FOR  LIQUIDS. 

Invention  of  pump,  252.  Reason  of  the  water  rising,  253.  Suction  pump,  254.  Effect 
of  untraversed  space,  255.  Force  necessary  to  raise  the  piston,  256.  Efficiency,  257. 
Forcing  pump,  258.  Plunger,  259.  Fire-engine,  260.  Double-acting  pumps,  261. 
Centrifugal  pumps,  262.  Jet-pump,  263.  Hydraulic  press,  264. 

CHAPTER  XXIII.    EFFLUX  OF  LIQUIDS. 

Torricelli's  theorem,  265.  Froude's  calculation  of  area  of  contracted  vein,  266.  Con- 
tracted vein  for  orifice  in  thin  plate,  267.  Apparatus  for  illustrating  Torricelli's 
theorem,  268.  Efflux  from  air-tight  space,  269.  Intermittent  fountain,  270. 
Siphon,  271.  Starting  the  Siphon,  272.  Siphon  for  sulphuric  acid,  273.  Tantalus' 
cup,  274.  Mariotte's  bottle,  275. 

EXAMPLES. 

Parallelogram  of  Velocities,  and  Parallelogram  of  Forces.     Ex.  1-11,  .     .     .  P239 

Parallel  Forces  and  Centre  of  Gravity.     Ex.  10*-33,     ....  !  239 

Work  and  Stability.     Ex.  34-43,   ........  "  241 

Inclined  Plane,  &c.     Ex.  44-48,      ..........  !  242 

Force,  Mass,  and  Velocity.     Ex.  49-59  ........  042 

Falling  Bodies  and  Projectiles.     Ex.  60-  83,      ....  '     243 

Atwood's  Machine.     Ex.  84-89  ........     '  '     244 

Energy  and  Work.     Ex.  90-98,      .......     *    '  '245 

Centrifugal  Force.     Ex.  99-101,     .......  '  '     245 

Pendulum,  and  Moment  of  Inertia.     Ex.  101*-107,       .     .  .     .     .     .     .          '.     246 

Pressure  of  Liquids.     Ex.  108-123,     .......  -  .  <,4g 

Density,  and  Principle  of  Archimedes.     Ex.  124-159  '     947 

Capillarity.     Ex.  160-164  ..........  '     ~  ' 

Barometer,  and  Boyle's  law.     Ex.  165-181 

Pumps,  &c.     Ex.  182-189,     ......     •    •    1    ."!."!.'.'  '     251 


AXSWCBS  TO  EXAMPLES, 


FRENCH  AND  ENGLISH  MEASURES. 


A  DECIMETRE   DIVIDED   INTO   CENTIMETRES  AND    MILLIMETRES. 


INCHES   AND  TENTHS. 


REDUCTION  OF  FRENCH  TO  ENGLISH  MEASURES. 


LENGTH. 

1  millimetre  =  '03937  inch,  or  about  -^  inch. 
1  centimetre^  '3937  inch. 
1  decimetre =3 -937  inch. 
1  metre =39 -37  inch =3  "281  ft.  =  l'0936  yd. 
1  kilometre =1093 '6  yds.,  or  about  £  mile. 
More  accurately,  1  metre =39 '370432  in. 
=3-2808693  ft.=l '09362311  yd. 

AREA. 

1  sq.  millim.  ='00155  sq.  in. 

1  sq.  centim.  =  *155  sq.  in. 

1  sq.  decim.  =15'5  sq.  in. 

1  sq.  metre  =  1550  sq.  in.  =  10764  sq.  ft.  = 

1-196  sq.  yd. 

VOLUME. 

1  cub.  millim.  =  '000061  cub.  in. 

1  cub.  centim.  = '061025  cub.  in. 

1  cub.  decim.  =61 '0254  cub.  in. 

cub.  metre=61025  cub.  in.  =35 '3156  cub. 

ft.  =  1'308  cub.  yd. 


The  Litre  (used  for  liquids)  is  the  same  as 
the  cubic  decimetre,  and  is  equal  to  1'7617 
pint,  or  -22021  gallon. 

MASS  AND  WEIGHT. 

1  milligramme  ='01543  grain. 

1  gramme         =15'432  grain. 

1  kilogramme  =  15432  grains=2'205  Ibs.  avoir. 

More  accurately,  the  kilogramme  is 

2-20402125  Ibs. 

MISCELLANEOUS. 

1  gramme  per  sq.  centim.      =2 '0481  Ibs.  per 

sq.  ft. 
1  kilogramme  per  sq.  centim.  =  14 -223  Ibs.  per 

sq.  in. 

1  kilogrammetre=7'2331  foot-pounds. 
1  force  de  cheval=75  kilogrammetres  per 
second,  or  542^  foot-pounds  per  second  nearly, 
whereas  1   horse-power  (English)=550  foot- 
pounds per  second. 


REDUCTION  TO  C.G.S.   MEASURES.     (See  page  48.) 
[cm.  denotes  centimetre  (s);  gm.  denotes  gramme(s).] 


LENGTH. 

1  inch.  =2-54  centimetres,  nearly. 

1  foot  =30-48  centimetres,  nearly. 

1  yard  =91  "44  centimetres,  nearly. 

1  statute  mile =160933  centimetres,  nearly. 
More  accurately,  1  inch=2'5399772  centi- 
metres. 

AREA. 

1  sq.  inch  =6-45  sq.  cm.,  nearly. 
1  sq.  foot  =929  sq.  cm.,  nearly. 
1  sq.  yard =8361  sq.  cm.,  nearly. 
1  sq.  mile=2'59xl010  sq.  cm.,  nearly. 

VOLUME. 

1  cub.  inch  =16'39  cub.  cm.,  nearly. 
1  cub.  foot  =23316  cub.  cm.,  nearly. 


1  cub.  yard= 764535  cub.  cm.,  nearly. 
1  gallon       =4541  cub.  cm.,  nearly. 

MASS. 

1  grain  =  -0648  gramme,  nearly. 
1  oz.  avoir.  =  28 '35  gramme,  nearly. 
1  Ib.  avoir.  =453-6  gramme,  nearly. 
1  ton  =  1  '016  x  106  gramme,  nearly. 

More  accurately,  1  Ib.  avoir.  =453 '59265  gm. 

VELOCITY. 

1  mile  per  hour         =447.04  cm.  per  seo, 
1  kilometre  per  hour =27  7  cm.  per  sec. 

DENSITY. 
1  Ib.  per  cub.  foot       =  '016019  gm,  per  cub. 

cm. 
62 '4  Ibs.  per  cub.  ft.    =1  gm.  per  cub.  cm. 


Xll 


FRENCH  AND   ENGLISH  MEASURES. 


FORCE  (assuming  # =981).     (See  p.  43.) 
Weight  of  1  grain      =  63  '57  dynes,  nearly. 

1  oz.  avoir.  =278  x  lOMynes.nearly. 
1  Ib.  avoir.  =  4 -45  x  105dynes,nearly. 
1  ton  =  9  '97  x  108  dynes,nearly. 
1  gramme —981  dynes,  nearly. 
1  kilogramme  =  9'81  x  105  dynes, 
nearly. 

WORK  (assuming^  =  981).    (See  p.  48.) 
1  foot-pound  =  1  '356  x  1 07  ergs,  nearly. 

1  kilogrammetrs      =  9'81  x  107  ergs,  nearly. 
Work  in  a  second ") 

by  one  theoretical  |>:=7-46xl09  ergs,  nearly, 
"'horse." 


STRESS  (assuming  <7=9S1). 

1  Ib.  per  sq.  ft.         —479  dynes  per  sq.  cin., 

nearly. 
1  Ib.  per  sq.  inch      =6'9xl04  dynes  per  sq. 

cm.,  nearly. 
1  kilog.  per  sq.  cm.  =9'81  x  103  dynes  per  sq. 

cm.,  nearly. 

7GO  mm.  of  mercury  at  0°C.  =  1  '014  x  106  dynes 
per  sq.  cm. ,  nearly. 

30  inches  of  mercury  at  0°  C.  =  1  "0163  x  lU6 

dynes  per  sq.  cm.,  nearly. 

1  inch  of  mercury  at  0°  C.  r=3'38S  x  104  dynes 

per  sq.  cm.,  nearly. 


TABLE  OF  DENSITIES,  IN  GRAMMES  PER  CUBIC  CENTIMETRE. 


LIQUIDS. 

Pure  water  at  4°  C.,    -  - 

Sea  water,  ordinary,    -  - 

Alcohol,  pure,    -    -    -  - 

proof  spirit,  -  - 


-  -    -  1-000 

-  -    -  1-026 

-  -    -  -791 
---  -91G 

Ether, -716 

Mercury  at  0°  C., 13 "596 

Naphtha, -848 


SOLIDS. 


7'8  to  8-4 

»»     wire, 8-54 

Bronze, 8'4 

Copper,  cast, 8'6 

»     sheet, 8-8 

„      hammered. 8'9 

Gold, 19  to  19-6 

Iron,  cast, 6-95  to  7 '3 

„    wrought, 7'6    to  7'8 

Lead, n.4 

Platinum, 21  to  22 

er. 10-5 

el, 7-8  to  7-9 

. 7-3  to  7-5 


Zinc, 6-8  to  7'2 

Ice, -92 

Basalt, 3-00 

Brick, 2  to  2-17 

Brickwork, 1«8 

Chalk, 1-8  to  2-8 

Clay, 1-92 

Glass,  crown,      - -.      2  "5 

„      flint, 3-0 

Quartz  (rock-crystal), 2'65 

Sand, 1-42 

Fir,  spruce, -43  to  '7 

Oak,  European, -69  to  '99 

Lignum-vitse, 'Co  to  1  '33 

Sulphur,  octahedral, 2'05 

„       prismatic, 1-93 


GASES,  at  0°  C.  and  a  pressure  of  a  million 
dynes  per  sq.  cm. 

Air.  dry, -0012759 

Oxygen, -0014107 

Nitrogen, -0012393 

Hydrogen, '00008837 

Carbonic  acid, '00]  9509 


ELEMENTARY   TREATISE 

ON 

NATUBAL    PHILOSOPHY. 


CHAPTER  L 

INTRODUCTORY. 

1.  Natural  Science,  in  the  widest  sense  of  the  term,  comprises  all 
the  phenomena  of   the  material  world.     In   so  far  as   it   merely 
describes  and  classifies  these  phenomena,  it  may  be  called  Natural 
History;  in  so  far  as  it  furnishes  accurate  quantitative  knowledge 
of   the  relations   between  causes  and  effects   it   is  called   Natural 
Philosophy.     Many   subjects   of   study  pass   through   the   natural 
history  stage  before  they  attain  the  natural  philosophy  stage;  the 
phenomena  being  observed  and  compared  for  many  years  before  the 
quantitative  laws  which  govern  them  are  disclosed. 

2.  There  are  two  extensive  groups  of  phenomena  which  are  con- 
ventionally excluded  from  the  domain  of  Natural  Philosophy,  and 
regarded  as  constituting  separate  branches  of  science  in  themselves; 
namely: — 

First.  Those  phenomena  which  depend  on  vital  forces;  such 
phenomena,  for  example,  as  the  growth  of  animals  and  plants. 
These  constitute  the  domain  of  Biology. 

Secondly.  Those  which  depend  on  elective  attractions  between 
the  atoms  of  particular  substances,  attractions  which  are  known  by 
the  name  of  chemical  affinities.  These  phenomena  are  relegated  to 
the  special  science  of  Chemistry. 

Again,  Astronomy,  which  treats  of  the  nature  and  movements  of 
the  heavenly  bodies,  is,  like  Chemistry,  so  vast  a  subject,  that  it 
forms  a  special  science  of  itself;  though  certain  general  laws,  which 
its  phenomena  exemplify,  are  still  included  in  the  study  of  Natural 
Philosophy. 


2  INTRODUCTORY. 

3.  Those  phenomena  which  specially  belong  to  the  domain  of 
Natural  Philosophy  are  called  physical;  and  Natural  Philosophy 
itself  is  called  Physics.  It  may  be  divided  into  the  following 
branches. 

I.  DYNAMICS,  or  the  general  laws  of  force  and  of  the  relations 
which  exist  between  force,  mass,  and  velocity.  These  laws  may  be 
applied  to  solids,  liquids,  or  gases.  Thus  we  have  the  three 
divisions,  Mechanics,  Hydrostatics,  and  Pneumatics. 

IT.  THERMICS;  the  science  of  Heat. 

III.  The  science  of  ELECTRICITY,  with  the  closely  related  subject 
of  MAGNETISM. 

IV.  ACOUSTICS;  the  science  of  Sound. 

V.  OPTICS;  the  science  of  Light. 

The  branches  here  numbered  I.  II.  III.  are  treated  in  Parts  I.  II. 

III.  respectively,  of  the  present  Work.    The  two  branches  numbered 

IV.  V.  are  treated  in  Part  IV. 


CHAPTER  II 


FIRST  PRINCIPLES   OF  DYNAMICS.      STATICS. 


4.  Force. — Force  may  be  defined  as  that  which  tends  to  produce 
motion  in  a  body  at  rest,  or  to  produce  change  of  motion  in  a  body 
which  is  moving.     A  particle  is  said  to  have  uniform  or  unchanged 
motion  when  it  moves  in  a  straight  line  with  constant  velocity;  and 
every  deviation  of  material  particles  from  uniform  motion  is  due  to 
forces  acting  upon  them. 

5.  Translation  and   Rotation. — When  a  body  moves  so  that  all 
lines  in  it  remain  constantly  parallel  to  their  original  positions  (or, 
to  use  the  ordinary  technical  phrase,  move  parallel  to  themselves), 
its  movement  is  called  a  pure  translation.     Since  the  lines  joining 
the  extremities  of  equal  and  parallel  straight  lines  are  themselves 
equal  and  parallel,  it  can  easily  be  shown  that,  in  such  motion,  all 
points  of  the  body  have  equal  and  parallel  velocities,  so  that  the 
movement  of  the  whole  body  is  completely  represented  by  the  move- 
ment of  any  one  of  its  points. 

On  the  other  hand,  if  one  point  of  a  rigid  body  be  fixed,  the  only 
movement  possible  for  the  body  is  pure  rotation,  the  axis  of  the 
rotation  at  any  moment  being  some  straight  line  passing  through 
this  point. 

Every  movement  of  a  rigid  body  can  be  specified  by  specifying 
the  movement  of  one  of  its  points  (any  point  will  do)  together  with 
the  rotation  of  the  body  about  this  point. 

6.  Force  which  acts  uniformly  on  all  the  particles  of  a  body,  as 
gravity  does  sensibly  in  the  case  of  bodies  of  moderate  size  on  the 
earth's  surface  (equal  particles  being  urged  with  equal  forces  and  in 
parallel   directions),  tends  to  give  the  body  a  movement  of  pure 
translation. 

In  elementary  statements  of  the  laws  of  force,  it  is  necessary,  for 


4  FIRST  PRINCIPLES   OF   DYNAMICS. 

the  sake  of  simplicity,  to  confine  attention  to  forces  tending  to 
produce  pure  translation. 

7.  Instruments  for  Measuring  Force.— We  obtain  the  idea  of  force 
through  our  own  conscious  exercise  of  muscular  force,  and  we  can 
approximately  estimate  the  amount  of  a  force  (if  not  too  great  or 
too  small)  by  the  effort  which  we  have  to  make  to  resist  it;  as  when 
we  try  the  weight  of  a  body  by  lifting  it. 

Dynamometers  are  instruments  in  which  force  is  measured  by 
means  of  its  effect  in  bending  or  otherwise  distorting  elastic  springs, 
and  the  spring-balance  is  a  dynamometer  applied  to  the  measure- 
ment of  weights,  the  spring  in  this  case  being  either  a  flat  spiral 
(like  the  mainspring  of  a  watch),  or  a  helix  (resembling  a  cork- 
screw). 

A  force  may  also  be  measured  by  causing  it  to  act  vertically 
downwards  upon  one  of  the  scale-pans  of  a  balance  and  counter- 
poising it  by  weights  in  the  other  pan. 

8.  Gravitation  Units  of  Force. — In  whatever  way  the  measurement 
of  a  force  is  effected,  the  result,  that  is,  the  magnitude  of  the  force, 
is  usually  stated  in  terms  of  weight;  for  example,  in  pounds  or  in 
kilogrammes.     Such  units  of  force  (called  gravitation  units)  are  to 
a  certain  extent  indefinite,  inasmuch  as  gravity  is  not  exactly  the 
same  over  the  whole  surface  of  the  earth;  but  they  are  sufficiently 
definite  for  ordinary  commercial  purposes. 

9.  Equilibrium,  Statics,  Kinetics. — When  a  body  free  to  move  is 
acted  on  by  forces  which  do  not  move  it,  these  forces  are  said  to  be 
in  equilibrium,  or  to  equilibrate  each  other.     They  may  equally 
well  be  described  as  balancing  each  other.     Dynamics  is  usually 
divided  into  two  branches.     The  first  branch,  called  Statics,  treats 
of    the    conditions    of    equilibrium.     The    second    branch,    called 
Kinetics,  treats  of  the  movements  produced  by  forces  not  in  equili- 
brium. 

10.  Action  and  Reaction. — Experiment  shows  that  force  is  always 
a  mutual  action  between  two  portions  of  matter.     When  a  body  is 
urged  by  a  force,  this  force  is  exerted  by  some  other  body,  which  is 
itself  urged  in  the  opposite  direction  with  an  equal  force.     When  I 
press  the  table  downwards  with  my  hand,  the  table  presses  my  hand 
upwards;  when  a  weight  hangs  by  a  cord  attached  to  a  beam,  the 
cord  serves  to  transmit  force  between  the  beam  and  the  weight,  so 
that,  by  the  instrumentality  of  the  cord,  the  beam  pulls  the  weight 
upwards  and  the  weight  pulls  the  beam  downwards.     Electricity 


EQUILIBRIUM   OF   TWO    FORGES.  5 

and  magnetism  furnish  no  exception  to  this  universal  law.  When 
a  magnet  attracts  a  piece  of  iron,  the  piece  of  iron  attracts  the 
magnet  with  a  precisely  equal  force. 

11.  Specification  of  a  Force  acting  at  a  Point. — Force  may  be 
applied  over  a  finite  area,  as  when  I  press  the  table  with  my  hand; 
or  may  be  applied  through  the  whole  substance  of  a  body,  as  in  the 
case  of  gravity;  but  it  is  usual  to  begin  by  discussing  the  action  of 
forces   applied   to  a  single  particle,  in  which  case  each   force   is 
supposed  to  act  along  a  mathematical  straight  line,  and  the  particle 
or  material  point  to  which  it  is  applied  is  called  its  point  of  applica- 
tion.    A  force  is  completely  specified  when  its  magnitude,  its  point 
of  application,  and  its  line  of  action  are  all  given. 

12.  Rigid   Body.     Fundamental   Problem  of  Statics.— A  force  of 
finite  magnitude  applied  to  a  mathematical  point  of  any  actual 
solid   body  would   inevitably   fracture   the   body.     To   avoid   this 
complication  and  other  complications  which  would  arise  from  the 
bending  and  yielding  of  bodies  under  the  action  of  forces,  the  fiction 
of  a  perfectly  rigid  body  is  introduced,  a  body  which  cannot  bend 
or  break  under  the  action  of  any  force  however  intense,  but  always 
retains  its  size  and  shape  unchanged. 

The  fundamental  problem  of  Statics  consists  in  determining  the 
conditions  which  forces  must  fulfil  in  order  that  they  may  be  in 
equilibrium  when  applied  to  a  rigid  body. 

13.  Conditions  of  Equilibrium  for  Two  Forces. — In  order  that  two 
forces   applied   to   a   rigid   body   should   be   in   equilibrium,  it   is 
necessary  and  sufficient  that  they  fulfil  the  following  conditions: — 

1st.  Their  lines  of  action  must  be  one  and  the  same. 

2nd.  The  forces  must  act  in  opposite  directions  along  this  common 
line. 

3rd.  They  must  be  equal  in  magnitude. 

It  will  be  observed  that  nothing  is  said  here  about  the  points  of 
application  of  the  forces.  They  may  in  fact  be  anywhere  upon  the 
common  line  of  action.  The  effect  of  a  force  upon  a  rigid  body  is 
not  altered  by  changing  its  point  of  application  to  any  other  point 
in  its  line  of  action.  This  is  called  the  principle  of  the  transmissi- 
bility  of  force. 

It  follows  from  this  principle  that  the  condition  of  equilibrium 
for  any  number  of  forces  with  the  same  line  of  action  is  simply  that 
the  sum  of  those  which  act  in  one  direction  shall  be  equal  to  the 
sum  of  those  which  act  in  the  opposite  direction. 


0  FIRST  PRINCIPLES   OF   DYNAMICS. 

H  Three  Forces  Meeting  in  a  Point.  Triangle  of  Forces.— If 
three  forces,  not  having  one  and  the  same  line  of  action  are  in 
equilibrium,  their  lines  of  action  must  lie  in  one  plane,  and  must 
either  meet  in  a  point  or  be  parallel.  We  shall  first  discuss  the  case 
in  which  they  meet  in  a  point. 

From  any  point  A  (Fig.  1)  draw  a  line  AB  parallel  to  one  ot  the 
two  given  forces,  and  so  that  in  travelling  from  A  to  B  we  should 
be  travelling  in  the  same  direction  in  which  the  force  acts  (not  m 
the  opposite  direction).     Also  let  it  be 
understood  that  the  length  of  AB  repre- 
sents the  magnitude  of  the  force. 

From  the  point  B  draw  a  line  BC 
representing  the  second  force  in  direc- 
tion, and  on  the  same  scale  of  magnitude 
on  which  AB  represents  the  first. 

Then  the  line  CA  will  represent  both 
in  direction  and  magnitude  the  third 
force  which  would  equilibrate  the  first 

Fig.  1. -Triangle  of  Forces. 

two. 

The  principle  embodied  in  this  construction  is  called  the  triangle 
of  forces.  It  may  be  briefly  stated  as  follows: — The  condition  of 
equilibrium  for  three  forces  acting  at  a  point  is,  that  they  be  repre- 
sented in  magnitude  and  direction  by  the  three  sides  of  a  triangle, 
taken  one  way  round.  The  meaning  of  the  words  "  taken  one  way 
round "  will  be  understood  from  an  inspection  of  the  arrows  with 
which  the  sides  of  the  triangle  in  Fig.  1  are  marked.  If  the 
directions  of  all  three  arrows  are  reversed  the  forces  represented 
will  still  be  in  equilibrium.  The  arrows  must  be  so  directed  that 
it  would  be  possible  to  travel  completely  round  the  triangle  by 
moving  along  the  sides  in  the  directions  indicated. 

When  a  line  is  used  to  represent  a  force,  it  is  always  necessary  to 
employ  an  arrow  or  some  other  mark  of  direction,  in  order  to  avoid 
ambiguity  between  the  direction  intended  and  its  opposite.  In  naming 
such  a  line  by  means  of  two  letters,  one  at  each  end  of  it,  the  order 
of  the  letters  should  indicate  the  direction  intended.  The  direction 
of  AB  is  from  A  to  B;  the  direction  of  BA  is  from  B  to  A. 

15.  Resultant  and  Components. — Since  two  forces  acting  at  a  point 
can  be  balanced  by  a  single  force,  it  is  obvious  that  they  are  equiv- 
alent to  a  single  force,  namely,  to  a  force  equal  and  opposite  to  that 
which  would  balance  them.  This  force  to  which  they  are  equivalent 


EQUILIBRIUM   OF   THREE  FORCES.  7 

is  called  their  resultant  Whenever  one  force  acting  on  a  rigid 
body  is  equivalent  to  two  or  more  forces,  it  is  called  their  resultant, 
and  they  are  called  its  components.  When  any  number  of  forces 
are  in  equilibrium,  a  force  equal  and  opposite  to  any  one  of  them  is 
the  resultant  of  all  the  rest. 

The  "triangle  of  forces"  gives  us  the  resultant  of  any  two  forces 
acting  at  a  point.  For  example,  in  Fig.  1,  AC  (with  the  arrow  in 
the  figure  reversed)  represents  the  resultant  of  the  forces  represented 
by  AB  and  BC. 

16.  Parallelogram  of  Forces. — The  proposition  called  the  "  parallel- 
ogram of  forces"  is  not  essentially  distinct  from  the  "triangle  of 
forces,"  but  merely  expresses  the  same  fact  from  a  slightly  different 
point  of  view.    It  is  as  follows: — If  two  forces 

acting  upon  the  same  rigid  body  in  lines 
which  meet  in  a  point,  be  represented  by  tivo 
lines  drawn  from  the  point,  and  a  parallelo- 
gram be  constructed  on  these  lines,  the  diagonal 
drawn  from  this  point  to  the  opposite  corner  FiS.  2. -Parallelogram  of 
of  the  parallelogram  represents  the  resultant 

For  example,  if  AB,  AC,  Fig.  2,  represent  the  two  forces,  AD  will 
represent  their  resultant. 

To  show  the  identity  of  this  proposition  with  the  triangle  of  forces, 
we  have  only  to  substitute  BD  for  AC  (which  is  equal  and  parallel 
to  it).  We  have  then  two  forces  represented  by  AB,  BD  (two  sides 
of  a  triangle)  giving  as  their  resultant  a  force  represented  by  the 
third  side  AD.  WTe  might  equally  well  have  employed  the  triangle 
ACD,  by  substituting  CD  for  AB. 

17.  Gravesande's  Apparatus. — An  apparatus  for  verifying  the  par- 
allelogram of  forces  is  represented  in  Fig.  3.    ACDB  is  a  light  frame 
in  the  form  of  a  parallelogram.    A  weight  P"  can  be  hung  at  A,  and 
weights  P,  F  can  be  attached,  by  means  of  cords  passing  over  pulleys, 
to  the  points  B,  C.     When  the  weights  P,  P',  F'  are  proportional  to 
AB,  AC  and  AD  respectively,  the  strings  attached  at  B  and  C  will 
be  observed  to  form  prolongations  of  the  sides,  and  the  diagonal  AD 
will  be  vertical. 

18.  Resultant  of  any  Number  of  Forces  at  a  Point. — To  find  the 
resultant  of  any  number  of  forces  whose  lines  of  action  meet  in  a 
point,  it   is   only  necessary  to  draw  a   crooked  line  composed  of 
straight  lines  which  represent  the  several  forces.     The  resultant  will 
be  represented  by  a  straight  line  drawn  from  the  beginning  to  the 


g  FIRST   PRINCIPLES   OF   DYNAMICS. 

end  of  this  crooked  line.  For  by  what  precedes,  if  ABODE  be  a 
rooked  line  such  that  the  straight  lines  AB  EC  CD,  ^  repent 
four  forces  acting  at  a  point,  we  may  substitute  for  AB  and  ] 


Fig.  3.— Gravesaiide's  Apparatus. 

the  straight  line  AC,  since  this  represents  their  resultant.  We  may 
then  substitute  AD  for  AC  and  CD,  and  finally  AE  for  AD  and  DE. 
One  of  the  most  important  applications  of 
this  construction  is  to  three  forces  not 
lying  on  one  plane.  In  this  case  the 
crooked  line  will  consist  of  three  edges  of 
a  parallelepiped,  and  the  line  which  repre- 
sents the  resultant  will  be  the  diagonal. 
This  is  evident  from  Fig.  4,  in  which  AB, 
AC,  AD  represent  three  forces  acting  at 
A.  The  resultant  of  AB  and  AC  is  Ar, 
and  the  resultant  of  Ar  and  AD  is  Ar'.  The  crooked  line  whose 
parts  represent  the  forces,  may  be  either  ABrr',  or  ABGr',  or  ADGv', 
&c.,  the  total  number  of  alternatives  being  six,  since  three  things 
can  be  taken  in  six  different  orders.  We  have  here  an  excellent 
illustration  of  the  fact  that  the  same  final  resultant  is  obtained, 
in  whatever  order  the  forces  are  combined 


Fig.  4.— Parallelepiped  of 
Forces. 


PARALLEL  FORCES.  9 

19.  Equilibrium  of  Three  Parallel  Forces. — If  three  parallel  forces, 
P,  Q,  R,  applied  to  a  rigid  body,  balance  each  other,  the  following 
conditions  must  be  fulfilled: —  Q 

1.  The  three  lines  of  action  AP,  BQ, 
CR,  Fig.  5,  must  be  in  one  plane. 

2.  The  two  outside  forces   P,  R,  must 
act    in    the    opposite    direction    to    the 
middle  force  Q,  and  their  sum  must  be 
equal  to  Q.  P 

3.  Each  force  must  be  proportional  to  Fig- 5- 

the  distance  between  the  lines  of  action  of  the  other  two;  that  is, 
we  must  have 

P  _  Q  _  R 
BC-AC-AB- 

The  figure  shows  that  AC  is  the  sum  of  AB  and  BC;  hence  it  fol- 
lows from  these  equations,  that  Q  is  equal  to  the  sum  of  P  and  R, 
as  above  stated. 

20.  Resultant  of  Two  Parallel  Forces. — Any  two  parallel  forces 
being  given,  a  third  parallel  force  which  will  balance  them  can  be 
found  from  the  above  equations;  and  a  force  equal  and  opposite  to 
this  will  be  their  resultant.     We  may  distinguish  two  cases. 

1.  Let  the  two  given  forces  be  in  the  same  direction.     Then  their 
resultant  is  equal  to  their  sum,  and  acts  in  the  same  direction,  along 
a  line  which  cuts  the  line  joining  their  points  of  application  into 
two  parts  which  are  inversely  as  the  forces. 

2.  Let  the  two  given  forces  be  in  opposite  directions.     Then  their 
resultant  will  be  equal  to  their  difference,  and  will  act  in  the  direc- 
tion of  the  greater  of  the  two  forces,  along  a  line  which  cuts  the 
production  of  the  line  joining  their  points  of  application  on  the  side 
of  the  greater  force;  and  the  distances  of  this  point  of  section  from 
the  two  given  points  of  application  are  inversely  as  the  forces. 

21.  Centre  of  Two  Parallel  Forces. — In  both  cases,  if  the  points  of 
application  are  not  given,  but  only  the  magnitudes  of  the  forces  and 
their  lines  of  action,  the  magnitude  and  line  of  action  of  the  resul- 
tant are  still  completely  determined;  for  all  straight  lines  which  are 
drawn  across  three  parallel  straight  lines  are  cut  by  them  in  the 
same  ratio;  and  we  shall  obtain  the  same  result  whatever  points  of 
application  we  assume. 

If  the  points  of  application  are  given,  the  resultant  cuts  the  line 


10  FIRST  PRINCIPLES   OF  DYNAMICS. 

joining  them,  or  this  line  produced,  in  a  definite  point,  whose  posi- 
tion depends  only  on  the  magnitudes  of  the  given  forces,  and  not  at 
all  on  the  angle  which  their  direction  makes  with  the  joining  line. 
This  result  is  important  in  connection  with  centres  of  gravity.  The 
point  so  determined  is  called  the  centre  of  the  two  parallel  forces. 
If  these  two  forces  are  the  weights  of  two  particles,  the  "centre" 
thus  found  is  their  centre  of  gravity,  and  the  resultant  force  is  the 
same  as  if  the  two  particles  were  collected  at  this  point. 

22.  Moments  of  Resultant  and  of  Components  Equal. — The  follow- 
ing proposition  is  often  useful.     Let  any  straight  line  be  drawn 
across  the  lines  of  action  of  two  parallel  forces  Plf  P2  (Fig.  6).     Let 

O  be  any  point  on  this  line,  and  xlt  x2 

0 -£± £ £*     the  distances  measured  from  0  to  the 

f  4-  i        points  of  section,  distances  measured 

2        in   opposite   directions   being   distin- 

Fig- 6-  guished  by  opposite  signs,  and  forces 

in  opposite  directions  being  also  distinguished  by  opposite  signs. 

Also  let  R  denote  the  resultant  of  Pj  and  P2,  and  x  the  distance 

from  0  to  its  intersection  with  the  line;  then  we  shall  have 

P!  xi  +  P2  x.t  -  R  X. 

For,  taking  the  standard  case,  as  represented  in  Fig.  G,  in  which  all 
the  quantities  are  positive,  we  have  OAX  =  xl}  OA2  =  x2,  OB  —  x, 
and  by  §  19  or  §  20  we  have 

Pi.A^Pa.BAj, 

that  is, 
whence 
that  is, 

Ri  =  Pla:1  +  P3a;>  (2) 

23.  Any  Number  of  Parallel  Forces  in  One  Plane.— Equation  (2) 
affords  the  readiest  means  of  determining  the  line  of  action  of  the 
resultant  of  several  parallel  forces  lying  in  one  plane.     For  let 
P1?  P2,  P8,  &c.,  be  the  forces,  R:  the  resultant  of  the  first  two  forces 
PL  P2,  and  R2  the  resultant  of  the  first  three  forces  Pl5  P2,  P3.     Let 
a  line  be  drawn  across  the  lines  of  action,  and  let  the  distances  of 
the  points  of  section  from  an  arbitrary  point  O  on  this  line  be 
expressed  according  to  the  following  scheme:— 

Force  PI         P2         P3     '  Rx         R2 

Distance  x,         x,        x3  x,         x 


MOMENT  OF  A  FORCE.  11 

Then,  by  equation  (2)  we  have 

E!  x1  =  Plx^  +  P2x.i. 

Also  since  R2  is  the  resultant  of  R:  and  P3,  we  have 

Ra^Raii  +  PaXs, 

and  substituting  for  the  term  Rx  Hcv  we  have 


This  reasoning  can  evidently  be  extended  to  any  number  of  forces, 
so  that  we  shall  have  finally 

KB  =  sum  of  such  terms  as  P#, 

where  R  denotes  the  resultant  of  all  the  forces,  and  is  equal  to  their 
algebraic  sum;  while  x  denotes  the  value  of  x  for  the  point  where 
the  line  of  action  of  R  cuts  the  fixed  line.  It  is  usual  to  employ  the 
Greek  letter  S  to  denote  "the  sum  of  such  terms  as."  We  may 

therefore  write 

R  =  2  (P) 
Kx=S  (Px) 

whence 

£=?-£*)  (3) 

S(P) 

24.  Moment  of  a  Force  about  a  Point.  —  When  the  fixed  line  is  at 
right  angles  to  the  parallel  forces,  the  product  7x  is  called  the 
moment  of  the  force  P  about  the  point  O.  More  generally,  the 
moment  of  a  force  about  a  point  is  the  product  of  the  force  by  the 
length  of  tJie  perpendicular  dropped  upon  it  from  the  point.  The 
above  equations  show  .that  for  parallel  forces  in  one  plane,  the 
moment  of  the  resultant  about  any  point  in  the  plane  is  the  sum  of 
the  moments  of  the  forces  about  the  same  point. 

If  the  resultant  passes  through  0,  the  distance  x  is  zero;  whence 
it  follows  from  the  equations  that  the  algebraical  sum  of  the 
moments  vanishes. 

The  moment  of  a  force  about  a  point  measures  the  tendency  of 
the  force  to  produce  rotation  about  the  point.  If  one  point  of  a 
body  be  fixed,  the  body  will  turn  in  one  direction  or  the  other 
according  as  the  resultant  passes  on  one  side  or  the  other  of  this 
point  (the  direction  of  the  resultant  being  supposed  given).  If  the 
resultant  passes  through  the  fixed  point,  the  body  will  be  in  equi- 
librium. 

The  moment  Px  of  any  force  about  a  point,  changes  sign  with  P 
and  also  with  x;  thereby  expressing  (what  is  obvious  in  itself)  that 


12 


FIRST   PRINCIPLES   OF   DYNAMICS. 


the  direction  in  which  the  force  tends  to  turn  the  body  about  the 
point  will  be  reversed  if  the  direction  of  P  is  reversed  while  its 
line  of  action  remains  unchanged,  and  will  also  be  reversed  if  the 
line  of  action  be  shifted  to  the  other  side  of  the  point  while  the 
direction  of  the  force  remains  unchanged. 

25    Experimental  Illustration.— Fig.  7  represents  a  simple  appar- 
atus (called  the  arithmetical  lever)  for  illustrating  the  laws  of  par- 


allel forces.  The  lever  AB  is  suspended  at  its  middle  point  by  a 
cord,  so  that  when  no  weights  are  attached  it  is  horizontal.  Equal 
weights  P,  P  are  hung  at  points  A  and  B  equidistant  from  the  centre, 
and  the  suspending  cord  after  being  passed  over  a  very  freely  mov- 
ing pulley  M,  has  a  weight  F  hung  at  its  other  end  sufficient  to  pro- 
duce equilibrium.  It  will  be  found  that  P'  is  equal  to  the  sum 
of  the  two  weights  P  together  with  the  weight  required  to  counter- 
poise the  lever  itself. 

In  the  second  figure,  the  two  weights  hung  from  the  lever  are  not 
equal,  but  one  of  them  is  double  of  the  other,  P  being  hung  at  B, 
and  2  P  at  C;  and  it  is  necessary  for  equilibrium  that  the  dis- 
tance OB  be  double  of  the  distance  OC.  The  weight  P'  required 


COUPLES.  13 

to  balance  the  system  will  now  be  3  P  together  with  the  weight 
of  the  lever. 

26.  Couple. — There  is  one  case  of  two  parallel  forces  in  opposite 
directions  which  requires  special  attention;  that  in  which  the  two 
forces  are  equal. 

To  obtain  some  idea  of  the  effect  of  two  such  forces,  let  us  first 
suppose  them  not  exactly  equal,  but  let  their  difference  be  very  small 
compared  with  either  of  the  forces.  In  this  case,  the  resultant  will 
be  equal  to  this  small  difference,  and  its  line  of  action  will  be  at  a 
great  distance  from  those  of  the  given  forces.  For  in  §  19  if  Q  is 
very  little  greater  than  P,  so  that  Q-P,  or  R  is  only  a  small  fraction 

of  P,  the  equation  ^=^  shows  that  AB  is  only  a  small  fraction 

of  BC,  or  in  other  words  that  BC  is  very  large  compared  with  AB. 

If  Q  gradually  diminishes  until  it  becomes  equal  to  P,  R  will 
gradually  diminish  to  zero;  but  while  it  diminishes,  the  product 
R .  BC  will  remain  constant,  being  always  equal  to  P .  AB. 

A  very  small  force  R  at  a  very  great  distance  would  have 
sensibly  the  same  moment  round  all  points  between  A  and  B  or 
anywhere  in  their  neighbourhood,  and  the  moment  of  R  is  always 
equal  to  the  algebraic  sum  of  the  moments  of  P  and  Q. 

When  Q  is  equal  to  P,  they  compose  what  is  called  a  couple,  and 
the  algebraic  sum  of  their  moments  about  any  point  in  their  plane 
is  constant,  being  always  equal  to  P .  AB,  which  is  therefore  called 
the  moment  of  the  couple. 

A  couple  consists  of  two  equal  and  parallel  forces  in  opposite 
directions  applied  to  the  same  body.  The  distance  between  their 
lines  of  action  is  called  the  arm  of  the  couple,  and  the  'product  of 
one  of  the  two  equal  forces  by  this  arm  is  called  the  moment  of  the 
couple. 

27.  Composition  of  Couples.     Axis  of  Couple. — A  couple  cannot  be 
balanced  by  a  single  force;  but  it  can  be  balanced  by  any  couple  of 
equal  moment,  opposite  in  sign,  if  the  plane  of  the  second  couple  be 
either  the  same  as  that  of  the  first  or  parallel  to  it. 

Any  number  of  couples  in  the  same  or  parallel  planes  are  equiva- 
lent to  a  single  couple  whose  moment  is  the  algebraic  sum  of  theirs. 

The  laws  of  the  composition  of  couples  (like  those  of  forces)  can 
be  illustrated  by  geometry. 

Let  a  couple  be  represented  by  a  line  perpendicular  to  its  plane, 
marked  with  an  arrow  according  to  the  convention  that  if  an 


14  FIRST  PRINCIPLES   OF  DYNAMICS. 

ordinary  screw  were  made  to  turn  in  the  direction  in  which  the 
couple  tends  to  turn,  it  would  advance  in  the  direction  in  which  the 
arrow  points.  Also  let  the  length  of  the  line  represent  the  moment 
of  the  couple.  Then  the  same  laws  of  composition  and  resolution 
which  hold  for  forces  acting  at  a  point  will  also  hold  for  couples. 
A  line  thus  drawn  to  represent  a  couple  is  called  the  axis  of  the 
couple. 

Just  as  any  number  of  forces  acting  at  a  point  are  either  in 
equilibrium  or  equivalent  to  a  single  force,  so  any  number  of  couples 
applied  to  the  same  rigid  body  (no  matter  to  what  parts  of  it)  are 
either  in  equilibrium  or  equivalent  to  a  single  couple. 

28.  Resultant  of  Force  and  Couple  in  Same  Plane. — The  resultant 
of  a  force  and  a  couple  in  the  same  plane  is  a  single  force.     For  the 

couple   may  be  replaced   by   another   of   equal 
moment  having  its  equal  forces  P,  Q,  each  equal 
1       to  the  given  force  F,  and  the  latter  couple  may 
**     then  be   turned    about    in   its   own  plane   and 
carried  into  such  a  position  that  one  of  its  two 
forces  destroys  the  force  F,  as  represented  in  Fig.  8.     There  will 
then  only  remain  the  force  P,  which  is  equal  and  parallel  to  F. 

By  reversing  this  procedure,  we  can  show  that  a  force  P  which 
does  not  pass  through  a  given  point  A  is  equivalent  to  an  equal  and 
parallel  force  F  which  does  pass  through  it,  together  with  a  couple; 
the  moment  of  the  couple  being  the  same  as  the  moment  of'  the  force 
P  about  A. 

29.  General  Resultant  of  any  Number  of  Forces  applied  to  a  Rigid 
Body. — Forces  applied  to  a  rigid  body  in  lines  which  do  not  meet 
in  one  point  are  not  in  general  equivalent  to  a  single  force.     By  the 
process   indicated    in  the  concluding    sentence   of    the   preceding 
section,  we  can  replace  the  forces  by  forces  equal  and  parallel  to 
them,  acting  at  any  assumed  point,  together  with  a  number  of 
couples.     These  couples  can  then  be  reduced  (by  the  principles  of 
§  27)  to  a  single  couple,  and  the  forces  at  the  point  can  be  replaced 
by  a  single  force;  so  that  we  shall  obtain,  as  the  complete  resultant, 
a  single  force  applied  at  any  point  we  choose  to  select,  and  a 
couple. 

We  can  in  general  make  the  couple  smaller  by  resolving  it  into 
two  components  whose  planes  are  respectively  perpendicular  and 
parallel  to  the  force,  and  then  compounding  one  of  these  components 
(the  latter)  with  the  force  as  explained  in  §  28,  thus  moving  the 


GENERAL  RESULTANT.  15 

force  parallel  to  itself  without  altering  its  magnitude.  This  is  the 
greatest  simplification  that  is  possible.  The  result  is  that  we  have 
a  single  force  and  a  couple  whose  plane  is  perpendicular  to  the 
force.  Any  combination  of  forces  that  can  be  applied  to  a  rigid 
body  is  reducible  to  a  force  acting  along  one  definite  line  and  a 
couple  in  a  plane  perpendicular  to  this  line.  Such  a  combination 
of  a  force  and  couple  is  called  a  wrench,  and  the  "  one  definite  line  " 
is  called  the  axis  of  the  wrench.  The  point  of  application  of  the 
force  is  not  definite,  but  is  any  point  of  the  axis. 

30.  Application  to  Action  and   Reaction. — Every  action  of  force 
that  one  body  can  exert  upon  another  is  reducible  to  a  wrench,  and 
the  law  of  reaction  is  that  the  second  body  will,  in  every  case,  exert 
upon  the  first  an  equal  and  opposite  wrench.     The  two  wrenches 
will  have  the  same  axis,  equal  and  opposite  forces  along  this  axis, 
and  equal  and  opposite  couples  in  planes  perpendicular  to  it. 

31.  Resolution  the  Inverse  of  Composition. — The  process  of  finding 
the  resultant  of  two  or  more   forces  is  called   composition.     The 
inverse  process  of  finding  two  or  more  forces  which  shall  together 
be  equivalent  to  a  given  force,  is  called  resolution,  and  the  two  or 
more  forces  thus  found  are  called  components. 

The  problem  to  resolve  a  force  into  two  components  along  two 
given  lines  which  meet  it  in  one  point  and  are  in  the  same  plane 
with  it,  has  a  definite  solution,  which  is  obtained  by  drawing  a 
triangle  whose  sides  are  parallel  respectively  to  the  given  force  and 
the  required  components.  The  given  force  and  the  required  com- 
ponents will  be  proportional  to  the  sides  of  this  triangle,  each  being 
represented  by  the  side  parallel  to  it. 

The  problem  to  resolve  a  force  into  three  components  along  three 
given  lines  which  meet  it  in  one  point  and  are  not  in  one  plane,  also 
admits  of  a  definite  solution. 

32.  Rectangular    Resolution. — In    the   majority  of   cases  which 
occur  in  practice  the  required  components  are  at  right  angles  to  each 
other,  and  the  resolution  is  then  said  to  be  rectangular.     When  "the 
component  of  a  force  along  a  given  line"  is  mentioned,  without 
anything  in  the  context   to   indicate  the  direction  of  the  other 
component  or  components,  it  is  always  to  be  understood  that  the 
resolution   is   rectangular.     The   process   of    finding    the   required 
component  in  this  case  is  illustrated  by  Fig.  9.    Let  AB  represent 
the  given  force  F,  and  let  AC  be  the  line  along  which  the  com- 
ponent of  F  is  required.     It  is  only  necessary  to  drop  from  B  a 


16  FIRST  PRINCIPLES   OF  DYNAMICS. 

perpendicular  BO  on  this  line;  AC  will  represent  the  required 
component.  CB  represents  the  other  component,  which,  along  with 
AC,  is  equivalent  to  the  given  force.  If 
the  total  number  of  rectangular  components, 
of  which  AC  represents  one,  is  to  be  three, 
[c  then  the  other  two  will  lie  in  a  plane  per- 


Fig.  o.-Component  along  a  given  pendicular  to  AC,  and  they  can  be  found  by 
again  resolving  CB.     The  magnitude  of  AC 

will  be  the  same  whether  the  number  of  components  be  two  or  three, 
and  the  c 
language, 


and  the  component  along  AC  will  be  F  ^~»  or  in  trigonometrical 


F  cos  .  BAG. 


We  have  thus  the  following  rule: — The  component  of  a  given  force 
along  a  given  line  is  found  by  multiplying  the  force  by  the  cosine 
of  the  angle  betiveen  its  own  direction  and  that  of  the  required 
component. 


CHAPTER    III. 


CENTRE   OF  GRAVITY. 


33.  Gravity  is  the  force  to  which  we  owe  the  names  "up"  and 
"  down."     The  direction  in  which  gravity  acts  at  any  place  is  called 
the  downward  direction,  and  a  line  drawn  accurately  in  this  direc- 
tion is  called  vertical;  it  is  the  direction  assumed  by  a  plumb-line. 
A  plane  perpendicular  to  this  direction  is  called  horizontal,  and  is 
parallel  to  the  surface  of  a  liquid  at  rest.     The  vertical  at  different 
places   are   not  parallel,  but   are   inclined   at   an   angle   which   is 
approximately  proportional    to   the   distance   between   the   places. 
It  amounts  to  180°  when  the  places  are  antipodal,  and  to  about  1' 
when  their  distance  is  one  geographical   mile,  or  to  about    1"  for 
every  hundred  feet.   ,  Hence,  when  we  are  dealing  with  the  action 
of  gravity  on  a  body  a  few  feet  or  a  few  hundred  feet  in  length, 
we   may   practically  regard   the   action   as   consisting   of   parallel 
forces. 

34.  Centre  of  Gravity. — Let  A  and  B  be  any  two  particles  of  a 
rigid  body,  let  wl  be  the  weight  of  the  particle  A,  and  w»  the  weight 
of  B.     These  weights  are  parallel  forces,  and  their  resultant  divides 
the  line  AB   in   the  inverse  ratio  of  the  forces.     As   the  body  is 
turned  about  into  different  positions,  the  forces  wl  and  w2  remain 
unchanged  in  magnitude,  and  hence  the  resultant  cuts  AB  always 
in  the  same  point.     This  point  is  called  the  centre  of  the  parallel 
forces  W-L  and  w<»  or  the  centre  of  gravity  of  the  two  particles  A  and 
B.     The  magnitude  of  the  resultant  will  be  w1-\-w2,  and  we  may 
substitute  it  for  the  two  forces  themselves;  in  other  words,  we  may 
suppose  the  two  particles  A  and  B  to  be  collected  at  their  centre 
of  gravity.     We  can  now  combine  this  resultant  with  the  weight 
of  a  third  particle  of  the  body,  and  shall  thus  obtain  a  resultant 

fr  passing  through  a  definite  point  in  the  line  which  joins 


J3  CENTRE  OF  GRAVITY. 

the  third  particle  to  the  centre  of  gravity  of  the  first  two.  The  first 
three  particles  may  now  be  supposed  to  be  collected  at  this  point, 
and  the  same  reasoning  may  be  extended  until  all  the  particles  have 
been  collected  at  one  point.  This  point  will  be  the  centre  of  gravity 
of  the  whole  body.  From  the  manner  in  which  it  has  been  ob- 
tained, it  possesses  the  property  that  the  resultant  of  all  the  forces 
of  gravity  on  the  body  passes  through  it,  in  every  position  in  which 
the  body  can  be  placed.  The  resultant  force  of  gravity  upon  a 
rigid  body  is  therefore  a  single  force  passing  through  its  centre 
of  gravity. 

35.  Centres  of  Gravity  of  Volumes,  Areas,  and  Lines.— If  the  body 
is  homogeneous  (that  is  composed  of  uniform  substance  throughout), 
the  position  of  the  centre  of  gravity  depends  only  on  the  figure,  and 
in  this  sense  it  is  usual  to  speak  of  the  centre  of  gravity  of  a  figure. 
In  like -manner  it  is  customary  to  speak  of  the  centres  of  gravity 
of  areas  and  lines,  an  area  being  identified  in  thought  with  a  thin 
uniform  plate,  and  a  line  with  a  thin  uniform  wire. 

It  is  not  necessary  that  a  body  should  be  rigid  in  order  that  it 
may  have  a  centre  of  gravity.  We  may  speak  of  the  centre  of 
gravity  of  a  mass  of  fluid,  or  of  the  centre  of  gravity  of  a  system 
of  bodies  not  connected  in  any  way.  The  same  point  which  would 
be  the  centre  of  gravity  if  all  the  parts  were  rigidly  connected,  is 
still  called  by  this  name  whether  they  are  connected  or  not. 

36.  Methods  of  Finding  Centres  of  Gravity. — Whenever  a  homo- 
geneous body  contains  a  point  which  bisects  all  lines  in  the  body 
that  can  be  drawn  through  it,  this   point  must   be  the  centre  of 
gravity.     The  centres  of  a  sphere,  a  circle,  a  cube,  a  square,  an 
ellipse,  an  ellipsoid,  a  parallelogram,  and  a  parallelepiped,  are  ex- 
amples. 

Again,  when  a  body  consists  of  a  finite  number  of  parts  whose 
weights  and  centres  of  gravity  are  known,  we  may  regard  each  part 
as  collected  at  its  own  centre  of  gravity. 

When  the  parts  are  at  all  numerous,  the  final  result  will  most 
readily  be  obtained  by  the  use  of  the  formula 

'=l%- 

where  P  denotes  the  weight  of  any  part,  x  the  distance  of  its  centre 
of  gravity  from  any  plane,  and  x  the  distance  of  the  centre  of 
gravity  of  the  whole  from  that  plane.  We  have  already  in  §  23 


CENTRE   OF   GRAVITY   OF  A   TRIANGLE.  19 

proved  this  formula  for  the  case  in  which  the  centres  of  gravity  lie 
in  one  straight  line  and  x  denotes  distance  from  a  point  in  this  line; 
and  it  is  not  difficult,  by  the  help  of  the  properties  of  similar 
triangles,  to  make  the  proof  general. 

37.  Centre  of  Gravity  of  a  Triangle. — To  find  the  centre  of  gravity 
of  a  triangle  ABC  (Fig.  10),  we  may  begin  by  supposing  it  divided 
into  narrow  strips  by  lines  (such  as  be)  parallel  to  EC.     It  can  be 
shown,  by  similar  triangles,  that  each  of  these  strips  is  bisected  by 
the   line   AD   drawn   from   A   to   D   the 

middle  point  of  BC.     But  each  strip  may 

be  collected  at  its  own  centre  of  gravity, 

that  is  at  its  own  middle  point;  hence  the 

whole  triangle  may  be  collected  on  the  line 

AD;  its  centre  of  gravity  must  therefore 

be  situated  upon  this  line.    Similar  reason-  -  D 

ing  shows  that  it  must  lie  upon  the  line  Fig  10- 

BE  drawn  from  B  to  the  middle  point  of  AC.     It  is  therefore  the 

intersection  of  these  two  lines.     If  we  join  DE  we  can  show  that 

the  triangles  AGB,  DGE,  are  similar,  and  that 

AG  _  AB 
GD~DE~ 

DG  is  therefore  one  third  of  DA.  The  centre  of  gravity  of  a 
triangle  therefore  lies  upon  the  line  joining  any  corner  to  the  middle 
point  of  the  opposite  side,  and  is  at  one-third  of  the  length  of  this 
line  from  the  end  where  it  meets  that  side. 

It  is  worthy  of  remark  that  if  three  equal  particles  are  placed  at 
the  corners  of  any  triangle,  they  have  the  same  centre  of  gravity  as 
the  triangle.  For  the  two  particles  at  B  and  C  may  be  collected  at 
the  middle  point  D,  and  this  double  particle  at  D,  together  with  the 
single  particle  at  A,  will  have  their  centre  of  gravity  at  G,  since  G 
divides  DA  in  the  ratio  of  1  to  2. 

38.  Centre  of  Gravity  of  a  Pyramid. — If  a  pyramid  or  a  cone  be 
divided  into  thin  slices  by  planes  parallel  to  its  base,  and  a  straight 
line  be  drawn  from  the  vertex  to  the  centre  of  gravity  of  the  base, 
this  line  will  pass  through  the  centres  of  gravity  of  all  the  slices, 
since  all  the  slices  are  similar  to  the  base,  and  are  similarly  cut  by 
this  line. 

In  a  tetrahedron  or  triangular  pyramid,  if  D  (Fig.  11)  be  the 
centre  of  gravity  of  one  face,  and  A  be  the  corner  opposite  to  this 


20  CENTRE   OF   GRAVITY. 

face,  the  centre  of  gravity  of  the  pyramid  must  lie  upon  the  line 
AD.  In  like  manner,  if  E  be  the 
centre  of  gravity  of  one  face,  the  centre 
of  gravity  of  the  pyramid  must  lie 
upon  the  line  joining  E  with  the  oppo- 
site corner  B.  It  must  therefore  be 
the  intersection  G  of  these  two  lines. 
That  they  do  intersect  is  otherwise 
obvious,  for  the  lines  AE,  BD  meet  in 
C,  the  middle  point  of  one  edge  of  the 
pyramid,  E  being  found  by  taking  CE 
-- -^  one  third  of  CA,  and  D  by  taking  CD 

Fig.  11.— Centre  of  Gravity  of  Tetrahedron.    one  third  of  CB. 

If  D,  E  be  joined,  we  can  show  that  the  joining  line  is  parallel  to 
BA,  and  that  the  triangles  AGB,  DGE  are  similar.     Hence 

AG  _  AB  _  BC 

GD  ~  DE  ~  DC  ~ 

That  is,  the  line  AD  joining  any  corner  to  the  centre  of  gravity  of 
the  opposite  face,  is  cut  in  the  ratio  of  3  to  1  by  the  centre  of  gravity 
G  of  the  triangle.  DG  is  therefore  one-fourth  of  DA,  and  the  dis- 
tance of  the  centre  of  gravity  from  any  face  is  one-fourth  of  the 
distance  of  the  opposite  corner. 

A  pyramid  standing  on  a  polygonal  base  can  be  cut  up  into  tri- 
angular pyramids  standing  on  the  triangular  bases  into  which  the 
polygon  can  be  divided,  and  having 
the  same  vertex  as  the  whole  pyramid. 
The  centres  of  gravity  of  these  trian- 
gular pyramids  are  all  at  the  same 
perpendicular  distance  from  the  base, 
namely  at  one-fourth  of  the  distance 
of  the  vertex,  which  is  therefore  the 
distance  of  the  centre  «of  gravity  of 
the  whole  from  the  base.  The  centre 
of  gravity  of  any  pyramid  is  there- 
fore found  by  joining  the  vertex  to 

Fig.  12.-Centre  of  Gravity  of  Pyramid,     tne  cent,re  Qf   gravity  of    the   base,  and 

cutting  off  one-fourth  of  the  joining  line  from  the  end  where  it  meets 
the  base.  The  same  rule  applies  to  a  cone,  since  a  cone  may  be 
regarded  as  a  polygonal  pyramid  with  a  very  large  number  of  sides. 


CENTRE  OF  GRAVITY  OF  PYRAMID. 


21 


Fig.  13. — Equilibrium  of  a  Body  supported  on  a  Horizontal 
Plane  at  three  or  more  Points. 


39.  If  four  equal  particles  are  placed  at  the  corners  of  a  triangular 
pyramid,  they  will  have  the  same  centre  of  gravity  as  the  pyramid. 
For  three  of  them  may,  as  we  have  seen  (§  37)  be  collected  at  the 
centre  of  gravity  of  one  face;  and  the  centre  of  gravity  of  the  four 
particles  will  divide  the  line  which  joins  this  point  to  the  fourth,  in 
the  ratio  of  1  to  3. 

40.  Condition  of  Standing  or  Falling.— When  a  heavy  body  stands 
on  a  base  of  finite  area, 

and  remains  in  equili- 
brium under  the  action 
of  its  own  weight  and  the 
reaction  of  this  base,  the 
vertical  through  its  centre 
of  gravity  must  fall  with- 
in the  base.  If  the  body 
is  supported  on  three  or 
more  points,  as  in  Fig.  13, 
we  are  to  understand  by 
the  base  the  convex1  poly- 
gon whose  corners  are  the 
points  of  support;  for  if  a  body  so  supported  turns  over,  it  must 
turn  about  the  line  joining  two  of  these  points. 

41.  Body  supported  at  one  Point. — When  a  heavy  body  supported 
at  one  point  remains  at  rest,  the  reaction  of  the  point  of  support 
equilibrates  the  force  of  gravity.     But  two  forces  cannot   be  in 
equilibrium  unless  they  have  the  same  line  of  action;  hence  the  ver- 
tical through  the  centre  of  gravity  of  the  body  must  pass  through 
the  point  of   support.     If   instead  of   being  supported  at  a  point, 
the  heavy  body  is  supported  by  an  axis  about  which  it  is  free  to 
turn,  the  vertical  through  the  centre  of  gravity  must  pass  through 
this  axis. 

42.  Stability  and  Instability.— When  the  point  of  support,  or  axis 
of  support,  is  vertically  below  the  centre  of  gravity,  it  is  easily  seen 
that,  if  the  body  were  displaced  a  little  to  either  side,  the  forces  act- 
ing upon  it  would  turn  it  still  further  away  from  the  position 
of  equilibrium.     On  the  other  hand,  when  the  point  or  axis  of  sup- 
port is  vertically  above  the  centre  of  gravity,  the  forces  which  would 

1  The  word  convex  is  inserted  to  indicate  that  there  must  be  no  re-entrant  angles. 
Any  points  of  support  which  lie  within  the  polygon  formed  by  joining  the  rest,  must  be 
left  out  of  account. 


22 


CENTRE   OF   GRAVITY. 


act  upon  it  if  it  were  slightly  displaced  would  tend  to  restore  it. 
In  the  latter  case  the  equilibrium  is  said  to  be  stable,  in  the  former 

unstable. 

When  the  centre  of  gravity  coincides  with  the  point  ot  support, 
or  lies  upon  the  axis  of  support,  the  body 
will  still  be  in  equilibrium  when  turned 
about  this  point  or  axis  into  any  other 
position.  In  this  case  the  equilibrium  is 
neither  stable  nor  unstable  but  is  called 
neutral. 

43.  Experimental  determination  of  Cen- 
tre of  Gravity. — In  general,  if  we  suspend 
a  body  by  any  point,  in  such  a  manner 
that  it  is  free  to  turn  about  this  point,  it 
will  come  to  rest  in  a  position  of  stable 
equilibrium.  The  centre  of  gravity  will 
then  be  vertically  beneath  the  point  of 

Fig.  14.-Experimental  Determination  Support.       If    W6    nOW    SUSpend    the    body 
of  Centre  of  Gravity.  fr()m  another   pointj  the  centre  Qf    gravity 

will  come  vertically  beneath  this.     The  intersection  of  these  two 
verticals  will  therefore  be  the  centre  of  gravity  (Fig.  14). 

44.  To  find  the  centre  of  gravity  of  a  flat  plate  or  board  (Fig.  15), 
we  may  suspend  it  from  a  point  near  its  circumfer- 
ence, in  such  a  manner  that  it  sets  itself  in  a  ver- 
tical plane.  Let  a  plumb-line  be  at  the  same  time 
suspended  from  the  same  point,  and  made  to  leave 
its  trace  upon  the  board  by  chalking  and  "snap- 
ping" it.  Let  the  board  now  be  suspended  from 
another  point,  and  the  operation  be  repeated.  The 
two  chalk  lines  will  intersect  each  other  at  that 
point  of  the  face  which  is  opposite  to  the  centre 
of  gravity;  the  centre  of  gravity  itself  being  of 
course  in  the  substance  of  the  board. 

45.  Work  done  against  Gravity.— When  a  heavy 
body  is  raised,  work  is  said  to  be  done  against  gravity,  and  the 
amount  of  this  work  is  reckoned  by  multiplying  together  the  weight 
of  the  body  and  the  height  through  which  it  is  raised.  Horizontal 
movement  does  not  count,  and  when  a  body  is  raised  obliquely,  only 
the  vertical  component  of  the  motion  is  to  be  reckoned. 

Suppose,  now,  that  we  have  a  number  of  particles  whose  weights 


Fig.  15.— Centre  of 
Gravity  of  Board. 


WORK  DONE   AGAINST   GRAVITY.  23 

are  ivl}  w2,  W3  &c.,  and  let  their  heights  above  a  given  horizontal 
plane  be  respectively  h1}  h2,  h3  &c.  We  know  by  equation  (3), 
§  23,  that  if  h  denote  the  height  of  their  centre  of  gravity  we 
have 

(wi  +  w.,  +  &c.)  K=W!  hi  +  w-i  A.J  +  &C.  (4) 

Let  the  particles  now  be  raised  into  new  positions  in  which  their 
heights  above  the  same  plane  of  reference  are  respectively  H1?  H2, 
H3  &c.  The  height  H  of  their  centre  of  gravity  will  now  be  such 
that 

(Wi  +  w.,,  +  &c.)  H  =  Wi  H!  +  Wi  H2  +  &c.  (5) 

From  these  two  equations,  we  find,  by  subtraction 

to  +  wj  +  fcc.)  (H-A)=«1  (H1-Al)  +  wJ  (H,-  A2)  +  &c.  (6) 

Now  H1  —  A!  is  the  height  through  which  the  particle  of  weight  u\ 
has  been  raised;  hence  the  work  done  against  gravity  in  raising  it  is 
10!  (Hj— Aj)  and  the  second  member  of  equation  (G)  therefore 
expresses  the  whole  amount  of  work  done  against  gravity.  But  the 
first  member  expresses  the  work  which  would  be  done  in  raising  all 
the  particles  through  a  uniform  height  H—  Ti,  which  is  the  height 
of  the  new  position  of  the  centre  of  gravity  above  the  old.  The 
work  done  against  gravity  in  raising  any  system  of  bodies  will 
therefore  be  correctly  computed  by  supposing  all  the  system  to  be 
collected  at  its  centre  of  gravity.  For  example,  the  work  done  in 
raising  bricks  and  mortar  from  the  ground  to  build  a  chimney,  is 
equal  to  the  total  weight  of  the  chimney  multiplied  by  the  height 
of  its  centre  of  gravity  above  the  ground. 

46.  The  Centre  of  Gravity  tends  to  Descend.— When  the  forces 
which  tend  to  move  a  system  are  simply  the  weights  of  its  parts,  we 
can  determine  whether  it  is  in  equilibrium  by  observing  the  path  in 
which  its  centre  of  gravity  would  travel  if  movement  took  place. 
If  we  suppose  this  path  to  represent  a  hard  frictionless  surface,  and 
the  centre  of  gravity  to  represent  a  heavy  particle  placed  upon  it, 
the  conditions  of  equilibrium  will  be  the  same  as  in  the  actual  case. 
The  centre  of  gravity  tends  to  run  down  hill,  just  as  a  heavy  particle 
does.  There  will  be  stable  equilibrium  if  the  centre  of  gravity  is  at 
the  bottom  of  a  valley  in  its  path,  and  unstable  equilibrium  if  it  is 
at  the  top  of  a  hill.  When  a  rigid  body  turns  about  a  horizontal 
axis,  the  path  of  its  centre  of  gravity  is  a  circle  in  a  vertical  plane. 
The  highest  and  lowest  points  of  this  circle  are  the  positions  of  the 
centre  of  gravity  in  unstable  and  stable  equilibrium  respectively; 


24  CENTKE  OF   GRAVITY. 

except  when  the  axis  traverses  the  centre  of  gravity  itself,  in  which 
case  the  centre  of  gravity  can  neither  rise  nor  fall,  and  the  equili- 
brium is  neutral. 

A  uniform  sphere  or  cylinder  lying  on  a  horizontal  plane  is  in 
neutral  equilibrium,  because  its  centre  of  gravity  will  neither  be 
raised  nor  lowered  by  rolling.  An  egg  balanced  on  its  end  as  in 
Fig.  16,  is  in  unstable  equilibrium,  because  its  centre  of  gravity  is  at 
the  top  of  a  hill  which  it  will  descend  when  the  egg  rolls  to  one  side. 
The  position  of  equilibrium  shown  in  Fig.  17  is  stable  as  regards 
rolling  to  left  or  right,  because  the  path  of  its  centre  of  gravity  in 


:J-  M 

, 


Fig  16.— Unstable  Equilibrium.  Fig.  17.— Stable  Equilibrium. 

such  rolling  would  be  a  curve  whose  lowest  point  is  that  now  occu- 
pied by  the  centre  of  gravity.  As  regards  rolling  in  the  direction  at 
right  angles  to  this,  if  the  egg  is  a  true  solid  of  resolution,  the  equili- 
brium is  neutral. 

47.  Work  done  by  Gravity. — When  a  heavy  body  is  lifted,  the 
lifting  force  does  work  against  gravity.     When  it  descends  gravity 
does  work  upon  it;  and  if  it  descends  to  the  same  position  from 
which  it  was  lifted,  the  work  done  by  gravity  in  the  descent  is 
equal  to  the  work  done  against  gravity  in  the  lifting;  each  being 
equal  to  the  weight  of  the  body  multiplied  by  the  vertical  displace- 
ment of  its  centre  of  gravity.     The  tendency  of  the  centre  of  gravity 
to  descend  is  a  manifestation  of  the  tendency  of  giavity  to  do  work; 
and  this  tendency  is  not  peculiar  to  gravity. 

48.  Work  done  by  any  Force.— A  force  is  said  to  do  work  when  its 
point  of  application  moves  in  the  direction  of  the  force,  or  in  any 
direction  making  an  acute  angle  with  this,  so  as  to  give  a  component 
displacement  in  the  direction  of  the  force;  and  the  amount  of  work 
done  is  the  product  of  the  force  by  this  component.     If  F  denote 


PRINCIPLE   OF   WORK.  25 

the  force,  a  the  displacement,  and  0  the  angle  between  the  two,  the 

work  done  by  F  is 

F  a  cos  0. 

which  is  what  we  obtain  either  by  the  above  rule  or  by  multiplying 
the  whole  displacement  by  the  effective  component  of  F,  that  is  the 
component  of  F  in  the  direction  of  the  displacement.  If  the  angle 
0  is  obtuse,  cos  0  is  negative  and  the  force  F  does  negative  work.  If 
0  is  a  right  angle  F  does  no  work.  In  this  case  F  neither  assists 
nor  resists  the  displacement.  When  0  is  acute,  F  assists  the  dis- 
placement, and  would  produce  it  if  the  body  were  constrained  by 
guides  which  left  it  free  to  take  this  displacement  and  the  directly 
opposite  one,  while  preventing  all  others. 

If  &  is  obtuse,  F  resists  the  displacement,  and  would  produce  the 
opposite  displacement  if  the  body  wrere  constrained  in  the  manner 
just  supposed. 

49.  Principle  of  Work. — If  any  number  of  forces  act  upon  a  body 
which  is  only  free  to  move  in  a  particular  direction  and  its  opposite, 
we  can  tell  in  which  of  these  two  directions  it  will  move  by  calcu- 
lating the  work  which  each  force  would  do.  Each  force  would  do 
positive  work  when  the  displacement  is  in  one  direction,  and  nega- 
tive work  when  it  is  in  the  opposite  direction,  the  absolute  amounts 
of  work  being  the  same  in  both  cases  if  the  displacements  are  equal. 
The  body  will  upon  the  whole  be  urged  in  that  direction  which  gives 
an  excess  of  positive  work  over  negative.  If  no  such  excess  exists, 
but  the  amounts  of  positive  and  negative  wrork  are  exactly  equal, 
the  body  is  in  equilibrium.  This  principle  (which  has  been  called 
the  principle  of  virtual  velocities,  but  is  better  called  the  principle 
of  work)  is  often  of  great  use  in  enabling  us  to  calculate  the  ratio 
which  two  forces  applied  in  given  wrays  to  the  same  body  must  have 
in  order  to  equilibrate  each  other.  It  applies  not  only  to  the 
"mechanical  powers"  and  all  combinations  of  solid  machinery,  but 
also  to  hydrostatic  arrangements;  for  example  to  the  hydraulic 
press.  The  condition  of  equilibrium  between  two  forces  applied  to 
any  frictionless  machine  and  tending  to  drive  it  opposite  ways,  is 
that  in  a  small  movement  of  the  machine  they  would  do  equal  and 
opposite  amounts  of  wrork.  Thus  in  the  screw-press  (Fig.  30)  the 
force  applied  to  one  of  the  handles,  multiplied  by  the  distance 
through  which  this  handle  moves,  will  be  equal  to  the  pressure 
which  this  force  produces  at  the  foot  of  the  screw,  multiplied  by  the 
distance  that  the  screw  travels. 


26  CENTRE   OF   GRAVITY. 

This  is  on  the  supposition  of  no  friction.  A  frictionless  machine 
gives  out  the  same  amount  of  work  which  is  spent  in  driving  it. 
The  effect  of  friction  is  to  make  the  work  given  out  less  than  the 
work  put  in.  Much  fruitless  ingenuity  has  been  expended  upon 
contrivances  for  circumventing  this  law  of  nature  and  producing  a 
machine  which  shall  give  out  more  work  than  is  put  into  it.  Such 
contrivances  are  called  "  perpetual  motions." 

50.  General  Criterion  of  Stability. — If  the  forces  which  act  upon 
a  body  and  produce  equilibrium  remain  unchanged  in  magnitude 
and  direction  when  the  body  moves  away  from  its  position,  and 
if  the  velocities  of  their  points  of  application  also  remain  unchanged 
in  direction  and  in  their  ratio  to  each  other,  it  is  obvious  that  the 
equality  of    positive  and  negative   work   which   subsists   at    the 
beginning  of  the  motion  will  continue  to  subsist  throughout  the 
entire   motion.     The   body  will   therefore  remain   in   equilibrium 
when  displaced.     Its  equilibrium  is  in  this  case  said  to  be  neutral. 

If  the  forces  which  are  in  equilibrium  in  a  given  position  of  the 
body,  gradually  change  in  direction  or  magnitude  as  the  body  moves 
away  from  this  position,  the  equality  of  positive  and  negative 
work  will  not  in  general  continue  to  subsist,  and  the  inequality  will 
increase  with  the  displacement.  If  the  body  be  displaced  with  a 
constant  velocity  and  in  a  uniform  manner,  the  rate  of  doing  work, 
which  is  zero  at  first,  will  not  continue  to  be  zero,  but  will  have  a 
value,  whether  positive  or  negative,  increasing  in  simple  proportion 
to  the  displacement.  Hence  it  can  be  shown  that  the  whole  work 
done  is  proportional  to  the  square  of  the  displacement,  for  when  we 
double  the  displacement  we,  at  the  same  time,  double  the  mean 
working  force. 

If  this  work  is  positive,  the  forces  assist  the  displacement  and  tend 
to  increase  it;  the  equilibrium  must 'therefore  have  been  unstable. 

On  the  other  hand,  if  the  work  is  negative  in  all  possible  displace- 
ments from  the  position  of  equilibrium,  the  forces  oppose  the 
displacements  and  the  equilibrium  is  stable. 

51.  Illustration  of  Stability.— A  good  example  of  stable  equili- 
brium of  this  kind  is  furnished  by  Gravesande's  apparatus  (Fig.  3) 
simplified  by  removing  the  parallelogram  and  employing  a  string 
to  support  the  three  weights,  one  of  them  P"  being  fastened  to  it  at 
a  point  A  near  its  middle,  and  the  others  P,  P'  to  its  ends.     The 
point  A  will  take  the  same  position  as  in  the  figure,  and  will  return 
to  it  again  when  displaced.     If  we  take  hold  of  the  point  A  and 


STABILITY.  27 

move  it  in  any  direction  whether  in  the  plane  of  the  string  or  out 
of  it,  we  feel  that  at  first  there  is  hardly  any  resistance  and  the 
smallest  force  we  can  apply  produces  a  sensible  disturbance;  but 
that  as  the  displacement  increases  the  resistance  becomes  greater. 
If  we  release  the  point  A  when  displaced,  it  will  execute  oscillations, 
which  will  become  gradually  smaller,  owing  to  friction,  and  it  will 
finally  come  to  rest  in  its  original  position  of  equilibrium. 

The  centre  of  gravity  of  the  three  weights  is  in  its  lowest 
position  when  the  system  is  in  equilibrium,  and  when  a  small  dis- 
placement is  produced  the  centre  of  gravity  rises  by  an  amount 
proportional  to  its  square,  so  that  a  double  displacement  produces 
a  quadruple  rise  of  the  centre  of  gravity. 

In  this  illustration  the  three  forces  remain  unchanged,  and  the 
directions  of  two  of  them  change  gradually  as  the  point  A  is  moved. 
Whenever  the  circumstances  of  stable  equilibrium  are  such  that  the 
forces  make  no  abrupt  changes  either  in  direction  or  magnitude  for 
small  displacements,  the  resistance  will,  as  in  this  case,  be  propor- 
tional to  the  displacement  (when  small),  and  the  work  to  the  square 
of  the  displacement,  and  the  system  will  oscillate  if  displaced  and 
then  left  to  itself. 

52.  Stability  where  Forces  vary  abruptly  with  Position. — There 
are  other  cases  of  stable  equilibrium  which  may  be  illustrated  by 
the  example  of  a  book  lying  on  a  table.     If  we  displace  it  by  lifting 
one  edge,  the  force  which  we  must  exert  does  not  increase  with  the 
displacement,  but  is  sensibly  constant  when  the  displacement  is 
small,  and  as  a  consequence  the  work  will  be  simply  proportional 
to  the  displacement.     The  reason  is,  that  one  of  the  forces  concerned 
in  producing  equilibrium,  namely,  the  upward  pressure  of  the  table, 
changes  per  saltum  at  the  moment  when  the  displacement  begins. 
In  applying  the  principle  of  work  to  such  a  case  as  this,  we  must 
employ,  instead  of  the  actual  work  done  by  the  force  which  changes 
abruptly,  the  work  which  it  would  do  if  its  magnitude  and  direction 
remained  unchanged,  or  only  changed  gradually. 

53.  Illustrations   from   Toys. — The   stability  of   the   "  balancer " 
(Fig.  18)  depends  on  the  fact  that,  owing  to  the  weight  of  the  two 
leaden  balls,  which  are  rigidly  attached  to  the  figure  by  stiff  wires, 
the  centre  of  gravity  of  the  whole  is  below  the  point  of  support. 
If  the  figure  be  disturbed  it  oscillates,  and  finally  comes  to  rest  in  a 
position  in  which  the  centre  of  gravity  is  vertically  under  the  toe 
on  which  the  figure  stands. 


28 


CENTRE   OF   GRAVITY. 


The  "tumbler"  (Fig.  19)  consists  of  a  light  figure  attached  to  a 
hemisphere   of    lead,   the   centre   of  gravity  of    the   whole  ^being 

between  the  centre  of  gravity  of 
the  hemisphere  and  the  centre  of 
the  sphere  to  which  it  belongs. 
When  placed  upon  a  level  table, 
the  lowest  position  of  the  centre 
of  gravity  is  that  in  which  the 
figure  is  upright,  and  it  accord- 
ingly returns  to  this  position  when 
displaced. 

54.  Limits  of  Stability. — In  the 
foregoing  discussion  we  have  em- 
ployed the  term  "stability"  in 
its  strict  mathematical  sense.  But 
there  are  cases  in  which,  though 
small  displacements  would  merely 
produce  small  oscillations,  larger 
displacements  would  cause  the 
body,  when  left  to  itself,  to  fall 
entirely  away  from  the  given 
position  of  equilibrium.  This  may 
Fig.  i8.-Baiancer.  be  expressed  by  saying  that  the 

equilibrium  is  stable  for  displacements  lying  within  certain  limits, 
but  unstable  for  displacements  beyond  these  limits.    The  equilibrium 


Fig  19.— Tur 


of  a  system  is  practically  unstable  when  the  displacements  which 
it  is  likely  to  receive  from  accidental  disturbances  lie  beyond  its 
limits  of  stability. 


CHAPTER  IV. 


THE  MECHANICAL   POWERS. 


55.  We  now  proceed  to  a  few  practical  applications  of  the  fore- 
going principles;  and  we  shall  begin  with  the  so-called  "mechanical 
powers,"  namely,  the  lever,  the  ivheel  and  axle,  the  pulley,  the 
inclined  plane,  the  wedge,  and  the  screw. 

56.  Lever. — Problems  relating  to  the  lever  are  usually  most  con- 
veniently  solved   by   taking   moments   round   the   fulcrum.      The 
general  condition  of  equilibrium  is,  that  the  moments  of  the  power 
and  the  weight  about  the  fulcrum  must  be  in  opposite  directions, 
and  must  be  equal.     When  the  power  and  weight  act  in  parallel 
directions,  the  conditions  of  equilibrium  are  precisely  those  of  three 
parallel   forces  (§  19),  the  third   force  being  the  reaction  of  the 
fulcrum. 

It  is  usual  to  distinguish  three  "  orders  "  of  lever.     In  levers  of 
the  first  order  (Fig.  20)  the  fulcrum  is  between  the  power  and  the 


hA  •* 

-o  o 


Fig.  20.  Fig.  21.  Fig.  22. 

Three  Orders  of  Lever. 

weight.  In  those  of  the  second  order  (Fig.  21)  the  weight  is 
between  the  power  and  the  fulcrum.  In  those  of  the  third  order 
(Fig.  22)  the  power  is  between  the  weight  and  the  fulcrum. 

In  levers  of  the  second  order  (supposing  the  forces  parallel),  the 
weight  is  equal  to  the  sum  of  the  power  and  the  pressure  on  the 
fulcrum;  and  in  levers  of  the  third  order,  the  power  is  equal  to 
the  sum  of  the  weight  and  the  pressure  on  the  fulcrum;  since 
the  middle  one  of  three  parallel  forces  in  equilibrium  must  always 
be  equal  to  the  sum  of  the  other  two. 


SO  THE  MECHANICAL   POWERS. 

57.  Arms.— The  arms  of  a  lever  are  the  two  portions  of  it  inter- 
mediate, respectively,  between  the  fulcrum  and   the   power,  and 
between  the  fulcrum  and  the  weight.     If  the  lever  is  bent,  or  if, 
though  straight,  it  is  not  at  right  angles  to  the  lines  of  action  of  the 
power  and  weight,  it  is  necessary  to  distinguish  between  the  arms 
of  the  lever  as  above  defined  (which  are  parts  of  the  lever),  and  the 
arms  of  the  power  and  weight  regarded  as   forces  which   have 
moments  round  the  fulcrum.     In  this  latter  sense  (which  is  always 
to  be  understood  unless  the  contrary  is  evidently  intended),  the 
arms  are  the  perpendiculars  dropped  from  the  fulcrum  upon  the 
lines  of  action  of  the  power  and  weight. 

58.  Weight  of  Lever. — In  the  above  statements  of  the  conditions 
of  equilibrium,  we  have  neglected  the  weight  of  the  lever  itself. 
To  take  this   into  account,  we  have  only  to  suppose  the  whole 
weight  of  the  lever  collected  at  its  centre  of  gravity,  and  then  take 
its  moment  round  the  fulcrum.     We  shall  thus  have  three  moments 
to  take  account  of,  and  the  sum  of  the  two  that  tend  to  turn  the 
lever  one  way,  must  be  equal  to  the  one  that  tends  to  turn  it  the 
opposite  way. 

59.  Mechanical  Advantage. — Every  machine  when  in  action  serves 
to  transmit  work  without  altering  its  amount;  but  the  force  which 
the  machine  gives  out  (equal  and  opposite  to  what   is  commonly 
called  the  weight)  may  be  much  greater  or  much  less  than  that  by 
which   it   is   driven   (commonly  called   the  power).     When   it   is 
greater,  the  machine  is  said  to  confer  mechanical  advantage,  and 
the  mechanical  advantage  is  measured  by  the  ratio  of  the  weight  to 
the  power  for  equilibrium.     In  the  lever,  when  the  power  has  a 
longer  arm  than  the  weight,  the  mechanical  advantage  is  equal  to 
the  quotient  of  the  longer  arm  by  the  shorter. 

60.  Wheel   and   Axle.— The   wheel  and  axle   (Fig.  23)   may  be 

regarded  as  an  endless  lever.  The  condition  of  equili- 
brium is  at  once  given  by  taking  moments  round  the 
common  axis  of  the  wheel  and  axle  (§  24).  If  we 
neglect  the  thickness  of  the  ropes,  the  condition  is  that 
the  power  multiplied  by  the  radius  of  the  wheel  must 
equal  the  weight  multiplied  by  the  radius  of  the  axle; 
but  it  is  more  exact  to  regard  the  lines  of  action  of  the 
two  forces  as  coinciding  with  the  axes  of  the  two  ropes, 
so  that  each  of  the  two  radii  should  be  increased  by  half  the  thick- 
ness of  its  own  rope.  If  we  neglect  the  thickness  of  the  ropes,  the 


PULLEYS. 


31 


mechanical  advantage  is  the  quotient  of  the  radius  of  the  wheel  by 
the  radius  of  the  axle. 

61.  Pulley. — A  pulley,  when  fixed  in  such  a  way  that  it  can  only 
turn  about  a  fixed  axis  (Fig.  24),  confers  no  mechanical  advantage. 
It  may  be  regarded  as  an  endless  lever  of  the  first  order  with  its 
two  arms  equal. 

The  arrangement  represented  in  Fig.  25  gives  a  mechanical 
advantage  of  2 ;  for  the  lower  or  movable  pulley  may  be  regarded 
as  an  endless  lever  of  the  second  order,  in  which  the  arm  of  the 
power  is  the  diameter  of  the  pulley,  and  the  arm  of  the  weight  is 


Fig.  24. 


Fig.  25. 


Fig.  -26. 


Fig.  2T. 


half  the  diameter.  The  same  result  is  obtained  by  employing  the 
principle  of  work;  for  if  the  weight  rises  1  inch,  2  inches  of  slack 
are  given  over,  and  therefore  the  power  descends  2  inches. 

62.  In  Fig.  26  there  are  six  pulleys,  three  at  the  upper  and  three 
at  the  lower  block,  and  one  cord  passes  round  them  all.     All  por- 
tions of  this  cord  (neglecting  friction)  are  stretched  with  the  same 
force,  which  is  equal  to  the  power;  and  six  of  these  portions,  parallel 
to  one  another,  support  the  weight.     The  power  is  therefore  one- 
sixth  of  the  weight,  or  the  mechanical  advantage  is  6. 

63.  In  the  arrangement  represented  in  Fig.  27,  there  are  three 
movable  pulleys,  each  hanging  by  a  separate  cord.     The  cord  which 
supports  the  lowest  pulley  is  stretched  with  a  force  equal  to  half 
the  weight,  since  its  two  parallel  portions  jointly  support  the  weight. 
The  cord  which  supports  the  next  pulley  is  stretched  with  a  force 
half  of  this,  or  a  quarter  of  the  weight;  and  the  next  cord  with  a 
force  half  of   this,  or  an  eighth  of  the  weight;   but  this  cord  is 
directly  attached  to  the  power.     Thus  the  power  is  an  eighth  of  the 


32  THE   MECHANICAL   POWERS. 

weight,  or  the  mechanical  advantage  is  8.  If  the  weight  and  the 
block1  to  which  it  is  attached  rise  1  inch,  the  next  block  rises  2 
inches,  the  next  4,  and  the  power  moves  through  8  inches.  Thus,  the 
work  done  by  the  power  is  equal  to  the  work  done  upon  the  weight. 
In  all  this  reasoning  we  neglect  the  weights  of  the  blocks  them- 
selves; but  it  is  not  difficult  to  take  them  into  account  when 
necessary. 

64.  Inclined   Plane. — We  now  come  to  the  inclined  plane.     Let 
AB  (Fio-.  28)  be  any  portion  of  such  a  plane,  and  let  AC  and  BC  be 

drawn  vertically  and  horizontally.     Then  AB 
t^-JLr  is  ca^d  tne  length,  AC   the  height,  and   CB 

/  /"-^-^^^     the  base  of  the  inclined  plane.     The  force  of 
c      ii  B  gravity  upon  a  heavy  body  M  resting  on  the 

N^p  plane,  may  be  represented  by  a  vertical  line 

Fis- 2S-  MP,  and  may  be  resolved  by  the  parallelogram 

of  forces  (§  1G)  into  two  components,  MT,  MN,  the  former  parallel 
and  the  latter  perpendicular  to  the  plane.  A  force  equal  and  oppo- 
site to  the  component  MT  will  suffice  to  prevent  the  body  from  slip- 
ping down  the  plane.  Hence,  if  the  power  act  parallel  to  the  plane, 
and  the  weight  be  that  of  a  heavy  body  resting  on  the  plane,  the 
power  is  to  the  weight  as  MT  to  MP;  but  the  two  triangles  MTP 
and  ACB  are  similar,  since  the  angles  at  M  and  A  are  equal,  and  the 
angles  at  T  and  C  are  right  angles;  hence  MT  is  to  MP  as  AC  to 
AB,  that  is,  as  the  height  to  the  length  of  the  plane. 

65.  The  investigation  is  rather  easier  by  the  principle  of  work 
(§  49).      The  work  done    by  the   power   in  drawing   the   heavy 
body  up   the   plane,   is   equal    to   the    power  multiplied    by  the 
length  of  the  plane.     But  the  work  done  upon  the  weight  is  equal 
to  the  weight  multiplied  by  the  height  through  which  it  is  raised, 
that  is,  by  the  height  of  the  plane.     Hence  we  have 

Power  x  length  of  plane  =  weight  X  height  of  plane;  or 
power  :  weight  :  :  height  of  plane  :  length  of  plane. 

66.  If,  instead  of  acting  parallel  to  the  plane,  the  power  acted 
parallel  to  the  base,  the  work  done  by  the  power  would   be  the 
product  of  the  power  by  the  base;  and  this  must  be  equal  to  the 
product  of  the  weight  by  the  height;  so  that  in  this  case  the  con- 
dition of  equilibrium  would  be — 

1  The  "  pulley  "  is  the  revolving  wheel.     The  pulley,  together  with  the  frame  in  which 
it  is  inclosed,  constitute  the  "  block." 


SCREW. 


33 


Power  :  weight  :  :  height  of  plane  :  base  of  plane. 

67.  Wedge. — In  these  investigations  we  have  neglected  friction. 
The  wedge  may  be  regarded  as  a  case  of  the  inclined  plane;  but  its 
practical  action  depends  to  such  a  large  extent  upon  friction  and 
impact1  that  we  cannot  profitably  discuss  it  here. 

68.  Screw. — The  screw  (Fig.  29)  is  also  a  case  of  the  inclined 
plane.     The  length  of  one  convolution  of  the  thread  is  the  length 
of  the  corresponding  inclined  plane,  the  step  of  the  screw,  or  distance 
between  two  successive  convolutions  (measured  parallel  to  the  axis 
of  the  screw),  is  the  height  of  the  plane,  and  the  circumference  of 


Fig  29. 


Fig.  30. 


the  screw  is  the  base  of  the  plane.  This  is  easily  shown  by  cutting 
out  a  right-angled  triangle  in  paper,  and  bending  it  in  cylindrical 
fashion  so  that  its  base  forms  a  circle. 

69.  Screw  Press. — In  the  screw  press  (Fig.  30)  the  screw  is  turned 
by  means  of  a  lever,  which  gives  a  great  increase  of  mechanical 
advantage.  In  one  complete  revolution,  the  pressures  applied  to  the 
two  handles  of  the  lever  to  turn  it,  do  work  equal  to  their  sum 
multiplied  by  the  circumference  of  the  circle  described  (approxi- 
mately) by  either  handle  (we  suppose  the  two  handles  to  be  equi- 
distant from  the  axis  of  revolution);  and  the  work  given  out  by  the 
machine,  supposing  the  resistance  at  its  lower  end  to  be  constant,  is 
equal  to  this  resistance  multiplied  by  the  distance  between  the 
threads.  These  two  products  must  be  equal,  friction  being  neglected. 

1  An  impact  (for  example  a  blow  of  a  hammer)  may  be  regarded  as  a  very  great  (and 
variable)  force  acting  for  a  very  short  time.  The  magnitude  of  an  impact  is  measured 
by  the  momentum  which  it  generates  in  the  body  struck. 


CHAPTEK    V, 


THE  BALANCE. 


70.  General  Description  of  the  Balance.— In  the  common  balance 
(Fig.  31)  there  is  a  stiff  piece  of  metal,  A  B,  called  the  beam,  which 

turns  about  the  sharp  edge 
O  of  a  steel  wedge  form- 
ing part  of  the  beam  and 
resting  upon  two  hard  and 
smooth  supports.  There  are 
two  other  steel  wedges  at 
A  and  B,  with  their  edges 
upwards,  and  upon  these 
edges  rest  the  hooks  for 
supporting  the  scale  pans. 
The  three  edges  (called 
knife-edges)  are  parallel  to 
one  another  and  perpen- 
I  dicular  to  the  length  of  the 
beam,  and  are  very  nearly 
in  one  plane. 

71.  Qualities   Requisite. — The  qualities   requisite   in   a   balance 
are: 

1.  That  it  be  consistent  with  itself;  that  is,  that  it  shall  give  the 
same  result  in  successive  weighings  of  the  same  body.     This  depends 
chiefly  on  the  trueness  of  the  knife-edges. 

2.  That  it  be  just.     This  requires  that  the  distances  A  O,  0  B,  be 
equal,  and  also  that  the  beam  remain  horizontal  when  the  pans  are 
empty.    Any  inequality  in  the  distances  A  0,  0  B,  can  be  detected 
by  putting  equal  (and  tolerably  heavy)  weights  into  the  two  pans. 
This  adds  equal  moments  if  the  distances  are  equal,  but  unequal 


Fig.  31  — Balance. 


SENSIBILITY   OF  BALANCE.  35 

moments  if  they  are  unequal,  and  the  greater  moment  will  prepon- 
derate. 

3.  Delicacy  or  sensibility  (that  is,  the  power  of  indicating  in- 
equality between  two  weights  even  when  their  difference  is  very 
small). 

This  requires  a  minimum  of  friction,  and  a  very  near  approach  to 
neutral  equilibrium  (§  40).  In  absolutely  neutral  equilibrium,  the 
smallest  conceivable  force  is  sufficient  to  produce  a  displacement  to 
the  full  limit  of  neutrality;  and  in  barely  stable  equilibrium  a  small 
force  produces  a  large  displacement.  The  condition  of  stability  is 
that  if  the  weights  supported  at  A  and  B  bo  supposed  collected  at 
these  edges,  the  centre  of  gravity  of  the  system  composed  of  the 
beam  and  these  two  weights  shall  be  below  the  middle  edge  O.  The 
equilibrium  would  be  neutral  if  this  centre  of  gravity  exactly  coin- 
cided with  O;  and  it  is  necessary  as  a  condition  of  delicacy  thnt  its 
distance  below  O  be  very  small. 

4.  Facility  for  weighing  quickly  is  desirable,  but  must  sometimes 
be  sacrificed  when  extreme  accuracy  is  required. 

The  delicate  balances  used  in  chemical  analysis  are  provided  with 
a  long  pointer  attached  to  the  beam.  The  end  of  this  pointer  moves 
along  a  graduated  arc  as  the  beam  vibrates;  and  if  the  weights  in  the 
two  pans  are  equal,  the  excursions  of  the  pointer  on  opposite  sides 
of  the  zero  point  of  this  arc  will  also  be  equal.  Much  time  is  con- 
sumed in  watching  these  vibrations,  as  they  are  very  slow;  and  the 
more  nearly  the  equilibrium  approaches  to  neutrality,  the  slower  they 
are.  Hence  quick  weighing  and  exact  weighing  are  to  a  certain  ex- 
tent incompatible. 

,  72.  Double  Weighing. — Even  if  a  balance  be  not  just,  yet  if  it  be 
consistent  with  itself,  a  correct  weighing  can  be  made  with  it  in  the 
following  manner: — Put  the  body  to  be  weighed  in  one  pan,  and 
counterbalance  it  with  sand  or  other  suitable  material  in  the  other. 
Then  remove  the  body  and  put  in  its  place  such  weights  as  are  just 
sufficient  to  counterpoise  the  sand.  These  weights  are  evidently 
equal  to  the  weight  of  the  body.  This  process  is  called  double 
weighing,  and  is  often  employed  (even  with  the  best  balances)  when 
the  greatest  possible  accuracy  is  desired. 

73.  Investigation  of  Sensibility.— Let  A  and  B  (Fig.  32)  be  the 
points  from  which  the  scale-pans  are  suspended,  0  the  axis  about 
which  the  beam  turns,  and  G  the  centre  of  gravity  of  the  beam.  If 
when  the  scale-pans  are  loaded  with  equal  weights,  we  put  into  one 


30 


THE   BALANCE. 


of  them  an  excess  of  weighty  the  beam  will  become  inclined  and 
will  take  a  position  such  as  A'B',  turning  through  an  angle  which 
we  will  call  a,  and  which  is  easily  calculated. 

In  fact  let  the  two  forces  P  and  P  +  p  act  at  A'  and  B  respec- 
tively where  P  denotes  the  less  of  the  two  weights,  including  the 

weight  of  the  pan.  Then  the  two 
forces  P  destroy  each  other  in  conse- 
quence of  the  resistance  of  the  axis 
O;  there  is  left  only  the  force  p 
applied  at  B',  and  the  weight  IT  of 
the  beam  applied  at  G',  the  new 
position  of  the  centre  of  gravity. 
p  These  two  forces  are  parallel,  and  are 
in  equilibrium  about  the  axis  0,  that 
is,  their  resultant  passes  through  the 
Fig.  32.  point  O.  The  distances  of  the  points 

of  application  of  the  forces  from  a  vertical  through  O  are  therefore 
inversely  proportional  to  the  forces  themselves,  which  gives  the 

relation 

TT.  G'K=p.  B'L. 

But  if  we  call  half  the  length  of  the  beam  I,  and  the  distance  OG  r 

we  have 

G'R=rsino,     B'L  =  1  cos  a. 

whence  vr  sin  <rrr  pi  cos  a,  and  consequently 


The  formula  (a)  contains  the  entire  theory  of  the  sensibility  of  the 
balance  when  properly  constructed.  We  see,  in  the  first  place,  that 
tan  a  increases  with  the  excess  of  weight  p,  which  was  evident  be- 
forehand. We  see  also  that  the  sensibility  increases  as  I  increases 
and  as  v  diminishes,  or,  in  other  words,  as  the  beam  becomes  longer 
and  lighter.  At  the  same  time  it  is  obviously  desirable  that,  under 
the  action  of  the  weights  employed,  the  beam  should  be  stiff  enough 
to  undergo  no  sensible  change  of  shape.  The  problem  of  the  balance 
then  consists  in  constructing  a  beam  of  the  greatest  possible  length 
and  lightness,  which  shall  be  capable  of  supporting  the  action  of 
given  forces  without  bending. 

Fortin,  whose  balances  are  justly  esteemed,  employed  for  his  beams 
bars  of  steel  placed  edgewise;  he  thus  obtained  great  rigidity,  but 


SENSIBILITY.  37 

certainly  not  all  the  lightness  possible.  At  present  the  makers  of 
balances  employ  in  preference  beams  of  copper  or  steel  made  in  the 
form  of  a  frame,  as  shown  in  Fig  33.  They  generally  give  them  the 
shape  of  a  very  elongated  lozenge,  the  sides  of  which  are  connected 
by  bars  variously  arranged.  The  determination  of  the  best  shape  is, 
in  fact,  a  special  problem,  and  is  an  application  on  a  small  scale  of 
that  principle  of  applied  mechanics  which  teaches  us  that  hollow 
pieces  have  greater  resisting  power  in  proportion  to  their  weight 
than  solid  pieces,  and  consequently,  for  equal  resisting  power,  the 
former  are  lighter  than  the  latter.  Aluminium,  which  with  a  rigidity 
nearly  equal  to  that  of  copper,  has  less  than  one-fourth  of  its  density, 
seems  naturally  marked  out  as  adapted  to  the  construction  of  beams. 
It  has  as  yet,  however,  been  little  used. 

The  formula  (a)  shows  us,  in  the  second  place,  that  the  sensibility 
increases  as  r  diminishes;  that  is,  as  the  centre  of  gravity  approaches 
the  centre  of  suspension.  These  two  points,  however,  must  not  coin- 
cide, for  in  that  case  for  any  excess  of  weight,  however  small,  the 
beam  would  deviate  from  the  horizontal  as  far  as  the  mechanism 
would  permit,  and  would  afford  no  indication  of  approach  to  equality 
in  the  weights.  With  equal  weights  it  would  remain  in  equilibrium 
in  any  position.  In  virtue  of  possessing  this  last  property,  such  a 
balance  is  called  indifferent.  Practically  the  distance  between  the 
centre  of  gravity  and  the  point  of  suspension  must  not  be  less  than 
a  certain  amount  depending  on  the  use  for  which  the  balance  is 
designed.  The  proper  distance  is  determined  by  observing  what 
difference  of  weights  corresponds  to  a  division  of  the  graduated  arc 
along  which  the  needle  moves.  If,  for  example,  there  are  20  divi- 
sions on  each  side  of  zero,  and  if  2  milligrammes  are  necessary  for 
the  total  displacement  of  the  needle,  each  division  will  correspond  to 
an  excess  of  weight  of  -^  or  TV  of  a  milligramme.  That  this  may 
be  the  case  we  must  evidently  have  a  suitable  value  of  r,  and  the 
maker  is  enabled  to  regulate  this  value  with  precision  by  means  of 
the  screw  which  is  shown  in  the  figure  above  the  beam,  and  which 
enables  him  slightly  to  vary  the  position  of  the  centre  of  gravity. 

74.  Weighing  with  Constant  Load. — In  the  above  analysis  we  have 
supposed  that  the  three  points  of  suspension  of  the  beam  and  of  the 
two  scale-pans  are  in  one  straight  line;  in  which  case  the  value  of 
tan  a  does  not  include  P,  that  is,  the  sensibility  is  independent  of  the 
weight  in  the  pans.  This  follows  from  the  fact  that  the  resultant 
of  the  two  forces  P  passes  through  0,  and  is  thus  destroyed,  because 

r;  fi  2  0  7 


gg  THE  BALANCE. 

the  axis  is  fixed.  This  would  not  be  the  case  if,  for  example,  the 
points  of  suspension  of  the  pans  were  above  that  of  the  beam;  in 
this  case  the  point  of  application  of  the  common  load  is  above  the 
point  O,  and,  when  the  beam  is  inclined,  acts  in  the  same  direction 
as  the  excess  of  weight;  whence  the  sensibility  increases  with  the 
load  up  to  a  certain  limit,  beyond  which  the  equilibrium  becomes 
unstable.1  On  the  other  hand,  when  the  points  of  suspension  of  the 
pans  are  below  that  of  the  beam,  the  sensibility  increases  as  the  load 
diminishes,  and,  as  the  centre  of  gravity  of  the  beam  may  in  this 
case  be  above  the  axis,  equilibrium  may  become  unstable  when  the 
load  is  less  than  a  certain  amount.  This  variation  of  the  sensibility 
with  the  load  is  a  serious  disadvantage;  for,  as  we  have  just  shown, 
the  displacement  of  the  needle  is  used  as  the  means  of  estimating 
weights,  and  for  this  purpose  we  must  have  the  same  displacement 
corresponding  to  the  same  excess  of  weight.  If  we  wish  to  employ 


I,    .....HI 

Fig.  33.— Beam  of  Balance. 

either  of  the  two  above  arrangements,  we  should  weigh  with  a  con- 
stant load.  The  method  of  doing  so,  which  constitutes  a  kind 
of  double  weighing,  consists  in  retaining  in  one  of  the  pans  a  weight 
equal  to  this  constant  load.  In  the  other  pan  is  placed  the  same 
load  subdivided  into  a  number  of  marked  weights.  When  the  body 

1  This  is  an  illustration  of  the  general  principle,  applicable  to  a  great  variety  of  philo- 
sophical apparatus,  that  a  maximum  of  sensibility  involves  a  minimum  of  stability ;  that 
is,  a  very  near  approach  to  instability.  This  near  approach  is  usually  indicated  by  exces- 
sive slowness  in  the  oscillations  which  take  place  about  the  position  of  equilibrium. 


BALANCES   OF  PRECISION. 


39 


to  be  weighed  is  placed  in  this  latter  pan,  we  must,  in  order  to  main- 
tain equilibrium,  remove  a  certain  number  of  weights,  which  evi- 
dently represent  the  weight  of  the  body. 

We  may  also  remark  that,  strictly  speaking,  the  sensibility  always 
depends  upon  the  load,  which  necessarily  produces  a  variation  in  the 
friction  of  the  axis  of  suspension.  Besides,  it  follows  from  the  nature 


Fig.  84. — Balance  for  Purposes  of  Accuracy. 

of  bodies  that  there  is  no  system  that  does  not  yield  somewhat  even 
to  the  most  feeble  action.  For  these  reasons,  there  is  a  decided 
advantage  in  operating  with  constant  load. 

75.  Details  of  Construction. — A  fundamental  condition  of  the  cor- 
rectness of  the  balance  is,  that  the  weight  of  each  pan  and  of  the 
load  which  it  contains  should  always  act  exactly  at  the  same  point, 
and  therefore  at  the  same  distance  from  the  axis  of  suspension. 
This  important  result  is  attained  by  different  methods.  The  arrange- 
ment represented  in  Fig.  33  is  one  of  the  most  effectual.  At  the 


40 


THE   BALANCE. 


extremities  of  the  beam  are  two  knife-edges,  parallel  to  the  axis  of 
rotation,  and  facing  upwards.  On  these  knife-edges  rests,  by  a 
hard  plane  surface  of  agate  or  steel,  a  stirrup,  the  front  of  which 
has  been  taken  away  in  the  figure.  On  the  lower  part  of  the  stirrup 
rests  another  knife-edge,  at  right  angles  to  the  former,  the  two  being 
together  equivalent  to  a  universal  joint  supporting  the  scale-pan  and 
its  contents.  By  this  arrangement,  whatever  may  be  the  position 
of  the  weights,  their  action  is  always  reduced  to  a  vertical  force  act- 
ing on  the  upper  knife-edge. 

Fig.  34  represents  a  balance  of  great  delicacy,  with  the  glass 
case  that  contains  it.  At  the  bottom  is  seen  the  extremity  of  a 
lever,  which  enables  us  to  raise  the  beam,  and  thus  avoid  wearing 
the  knife-edge  when  not  in  use.  At  the  top  may  be  remarked  an 
arrangement  employed  by  some  makers,  consisting  of  a  horizontal 
graduated  circle,  on  which  a  small  metallic  index  can  be  made  to 
travel;  its  different  displacements,  whose  value  can  be  determined 
once  for  all,  are  used  for  the  final  adjustment  to  produce  exact 
equilibrium. 

76.  Steelyard. — The  steelyard  (Fig.  35)  is  an  instrument  for 
weighing  bodies  by  means  of  a  single  weight,  P,  which  can  be  hung 

at  any  point  of  a 
graduated  arm  OB. 
As  P  is  moved  further 
from  the  fulcrum  O, 
its  moment  round  O 
increases,  and  there- 
fore the  weight  which 
must  be  hung  from 
the  fixed  point  A  to 
counterbalance  it  in- 
creases. Moreover, 
equal  movements  of 
P  along  the  arm  pro- 
duce equal  additions 
to  its  moment,  and  equal  additions  to  the  weight  at  A  produce 
equal  additions  to  the  opposing  moment.  Hence  the  divisions 
on  the  arm  (which  indicate  the  weight  in  the  pan  at  A)  must  be 
equidistant. 


Fig.  35. 


CHAPTER    VI 


FIRST   PRINCIPLES   OF   KINETICS. 


77.  Principle  of  Inertia. — A  body  not  acted  on  by  any  forces,  or 
only  acted  on  by  forces  which  are  in  equilibrium,  will  not  commence 
to  move;  and  if  it  be  already  in  motion  with  a  movement  of  pure 
translation,  it  will  continue  its  velocity  of  translation  unchanged,  so 
that  each  of  its  points  will  move  in  a  straight  line  with  uniform 
velocity.  This  is  Newton's  first  law  of  motion,  and  is  stated  by  him 
in  the  following  terms: — 

"  Every  body  continues  in  its  state  of  rest  or  of  uniform  motion 
in  a  straight  line,  except  in  so  far  as  it  is  compelled  by  impressed 
forces  to  change  that  state." 

The  tendency  to  continue  in  a  state  of  rest  is  manifest  to  the  most 
superficial  observation.  The  tendency  to  continue  in  a  state  of 
uniform  motion  can  be  clearly  understood  from  an  attentive  study  of 
facts.  If,  for  example,  we  make  a  pendulum  oscillate,  the  amplitude 
of  the  oscillations  slowly  decreases  and  at  last  vanishes  altogether. 
This  is  because  the  pendulum  experiences  resistance  from  the  air 
which  it  continually  displaces;  and  because  the  axis  of  suspension 
rubs  on  its  supports.  These  two  circumstances  combine  to  produce 
a  diminution  in  the  velocity  of  the  apparatus  until  it  is  completely 
annihilated.  If  the  friction  at  the  point  of  suspension  is  diminished 
by  suitable  means,  and  the  apparatus  is  made  to  oscillate  in  racuo, 
the  duration  of  the  motion  will  be  immensely  increased. 

Analogy  evidently  indicates  that  if  it  were  possible  to  suppress 
entirely  these  two  causes  of  the  destruction  of  the  pendulum's  velo- 
city, its  motion  would  continue  for  an  indefinite  time  unchanged. 

This  tendency  to  continue  in  motion  is  the  cause  of  the  effects 
which  are  produced  when  a  carriage  or  railway  train  is  suddenly 
stopped.  The  passengers  are  thrown  in  the  direction  of  the  motion, 


42  FIRST   PRINCIPLES   OF   KINETICS. 

in  virtue  of  the  velocity  which  they  possessed  at  the  moment  when 
the  stoppage  occurred.  If  it  were  possible  to  find  a  brake  sufficiently 
powerful  to  stop  a  train  suddenly  at  full  speed,  the  effects  of  such  a 
stoppage  would  be  similar  to  the  effects  of  a  collision. 

Inertia  is  also  the  cause  of  the  severe  falls  which  are  often  received 
in  alighting  incautiously  from  a  carriage  in  motion;  all  the  particles 
of  the  body  have  a  forward  motion,  and  the  feet  alone  being  reduced 
to  rest,  the  upper  portion  of  the  body  continues  to  move,  and  is  thus 
thrown  forward. 

When  we  fix  the  head  of  a  hammer  on  the  handle  by  striking  the 
end  of  the  handle  on  the  ground,  we  utilize  the  inertia  of  matter. 
The  handle  is  suddenly  stopped  by  the  collision,  and  the  head  con- 
tinues to  move  for  a  short  distance  in  spite  of  the  powerful  resist- 
ances which  oppose  it. 

78.  Second  Law  of  Motion.  —  Newton's  second  law  of  motion  is 
that  "  Change  of  motion  is  proportional  to  the  impressed  force  and 
is  in  the  direction  of  that  force." 

Change  of  motion  is  here  spoken  of  as  a  quantity,  and  as  a  directed 
quantity.  In  order  to  understand  how  to  estimate  change  of  motion, 
we  must  in  the  first  place  understand  how  to  compound  motions. 

When  a  boat  is  sailing  on  a  river,  the  motion  of  the  boat  relative 
to  the  shore  is  compounded  of  its  motion  relative  to  the  water  and 
the  motion  of  the  water  relative  to  the  shore.  If  a  person  is  walk- 
ing along  the  deck  of  the  boat  in  any  direction,  his  motion  relative 
to  the  shore  is  compounded  of  three  motions,  namely  the  two  above 
mentioned  and  his  motion  relative  to  the  boat. 

Let  A,  B  and  C  be  any  three  bodies  or  systems.'  The  motion  of 
A  relative  to  B,  compounded  with  the  motion  of  B  relative  to  C,  is 
the  motion  of  A  relative  to  C.  This  is  to  be  taken  as  the  definition 

of  what  is  meant 
by  compounding 
two  motions;  and  it 
leads  very  directly 

Fig.  36.-Composition  of  Motion,.  t()    th 


two  rectilinear  motions  are  compounded  by  the  parallelogram  law. 
For  if  a  body  moves  along  the  deck  of  a  ship  from  O  to  A  (Fig.  36), 
and  the  ship  in  the  meantime  advances  through  the  distance  OB,  it 
is  obvious  that,  if  we  complete  the  parallelogram  OBCA,  the  point 
A  of  the  ship  will  be  brought  to  C,  and  the  movement  of  the  body 
in  space  will  be  from  0  to  C.  If  the  motion  along  OA  is  uniform. 


SECOND   LAW   OF   MOTION.  43 

and  the  motion  of  the  ship  is  also  uniform,  the  motion  of  the 
body  through  space  will  be  a  uniform  motion  along  the  diagonal 
OC.  Hence,  if  two  component  velocities  be  represented  by  two  lines 
drawn  from  a  point,  and  a  parallelogram  be  constructed  on  these 
lines,  its  diagonal  will  represent  the  resultant  'velocity. 

It  is  obvious  that  if  OA  in  the  figure  represented  the  velocity  of 
the  ship  and  OB  the  velocity  of  the  body  relative  to  the  ship,  we 
should  obtain  the  same  resultant  velocity  OC.  This  is  a  general 
law;  the  interchanging  of  velocities  which  are  to  be  compounded 
does  not  affect  their  resultant. 

Now  suppose  the  velocity  OB  to  be  changed  into  the  velocity  OC, 
what  are  we  to  regard  as  the  change  of  velocity?  The  change  of 
velocity  is  that  velocity  Avhich  compounded  with  OB  would  give  OC 
It  is  therefore  OA.  The  same  force  which,  in  a  given  time,  acting 
always  parallel  to  itself,  changes  the  velocity  of  a  body  from  OB  to 
OC,  would  give  the  body  the  velocity  OA  if  applied  to  it  for  the  same 
time  commencing  from  rest.  Change  of  motion,  estimated  in  this 
way,  depends  only  on  the  acting  force  and  the  body  acted  on  by  the 
force;  it  is  entirely  independent  of  any  previous  motion  which  the 
body  may  possess.  No  experiments  on  forces  and  motions  inside  a 
carriage  or  steamboat  which  is  travelling  with  perfect  smoothness  in 
a  straight  course,  will  enable  us  to  detect  that  it  is  travelling  at  allt 
We  cannot  even  assert  that  there  is  any  such  thing  as  absolute  rest, 
or  that  there  is  any  difference  between  absolute  rest  and  uniform 
straight  movement  of  translation. 

As  change  of  motion  is  independent  of  the  initial  condition  of  rest 
or  motion,  so  also  is  the  change  of  motion  produced  by  one  force  act- 
ing on  a  body  independent  of  the  change  produced  by  any  other 
force  acting  on  the  body,  provided  that  each  force  remains  constant 
in  magnitude  and  direction.  The  actual  motion  will  be  that  which 
is  compounded  of  the  initial  motion  and  the  motions  due  to  the  two 
forces  considered  separately.  If  AB  represent  one  of  these  motions, 
BC  another,  and  CD  the  third,  the  actual  or  resultant  motion  will  be 
AD. 

The  change  produced  in  the  motion  of  a  body  by  two  forces  act- 
ing-jointly  can  therefore  be  found  by  compounding  the  changes 
which  would  be  produced  by  each  force  separately.  This  leads  at 
once  to  the  "  parallelogram  of  forces,"  since  the  changes  of  motion 
produced  in  one  and  the  same  body  are  proportional  to  the  forces 
which  produce  them,  and  are  in  the  directions  of  these  forces. 


44  FIRST   PRINCIPLES   OF   KINETICS. 

In  case  any  student  should  be  troubled  by  doubt  as  to  whether 
the  "  changes  of  motion"  which  are  proportional  to  the  forces,  are  to 
be  understood' as  distances,  or  as  velocities,  we  may  remark  that  the 
law  is  equally  true  for  both,  and  its  truth  for  one  implies  its  truth 
for  the  other,  as  will  appear  hereafter  from  comparing  the  formula 
for  the  distance  s  —  \j&,  with  the  formula  for  the  velocity  v  =  ft, 
since  both  of  these  expressions  are  proportional  to  /. 

79.  Explanation  of  Second  Law  continued. — It  is  convenient  to 
distinguish  between  the  intensity  of  a  force  and  the  magnitude  or 
amount  of  a  force.     The  intensity  of  a  force  is  measured  by  the 
change  of  velocity  which  the  force  produces  during  the  unit  of  time; 
and  can  be  computed  from  knowing  the  motion  of  the  body  acted 
on,  without  knowing  anything  as  to  its  mass.     Two  bodies  are  said 
to  be  of  equal  mass  when  the  same  change  of  motion  (whether  as 
regards  velocity  or  distance)  which  is  produced  by  applying  a  given 
force  to  one  of  them  for  a  given  time,  would  also  be  produced  by 
applying  this  force  to  the  other  for  an  equal  time.     If  we  join  two 
such  bodies,  we  obtain  a  body  of  double  the  mass  of  either;  and  if  we 
apply  the  same  force  as  before  for  the  same  time  to  this  double  mass, 
we  shall  obtain  only  half  the  change  of  velocity  or  distance  that  we 
obtained  before.    Masses  can  therefore  be  compared  by  taking  the  in- 

•  verse  ratio  of  the  changes  produced  in  their  velocities  by  equal  forces. 
The  velocity  of  a  body  multiplied  by  its  mass  is  called  the  momen- 
tum of  the  body,  and  is  to  be  regarded  as  a  directed  magnitude  hav- 
ing the  same  direction  as  the  velocity.  The  change  of  velocity,  when 
multiplied  by  the  mass  of  the  body,  gives  the  change  of  momentum ; 
and  the  second  law  of  motion  may  be  thus  stated: — 

The  change  of  momentum  ^oduced  in  a  given  time  is  jrropor- 
tional  to  the  force  which  produces  it,  and  is  in  the  direction  of  this 
force.  It  is  independent  of  the  mass;  the  change  of  velocity  in  a 
given  time  being  inversely  as  the  mass. 

80.  Proper  Selection  of  Unit  of  Force.— If  we  make  a  proper  selec- 
tion of  units,  the  change  of  momentum  produced  in  unit  time  will 
be  not  only  proportional  but  numerically  equal  to  the  force  which 
produces  it;  and  the  change  of  momentum  produced  in  any  time  will 
be  the  product  of  the  force  by  the  time.     Suppose  any  units  of 
length,  time,  and  mass  respectively  to  have  been  selected.     Then  the 
unit  velocity  will  naturally  be  denned  as  the  velocity  with  which 
unit  length  would  be  passed  over  in  unit  time;  the  unit  momentum 
will  be  the  momentum  of  the  unit  mass  moving  with  this  velocity; 


UNIT   OF   FORCE.  45 

and  the  unit  force  will  be  that  force  which  produces  this  momentum 
in  unit  time.  We  define  the  unit  force,  then,  as  that  force  which 
acting  for  unit  time  upon  unit  mass  produces  unit  velocity. 

81.  Relation  between  Mass  and  Weight. — The  weight  of  a  body, 
strictly  speaking,  is  the  force  with  which  the  body  tends  towards 
the  earth.     This  force  depends  partly  on  the  body  and  partly  on  the 
earth.     It  is  not  exactly  the  same  for  one  and  the  same  body  at  all 
parts  of  the  earth's  surface,  but  is  decidedly  greater  in  the  polar  than 
in  the  equatorial  regions.     Bodies  which,  when  weighed  in  a  balance 
in  vacuo,  counterbalance  each  other,  or  counterbalance  one  and  the 
same  third  body,  have  equal  weights  at  that  place,  and  will  also  be 
found  to  have  equal  weights  at  any  other  place.     Experiments  which 
we  shall  hereafter  describe  (§  89)  show  that  such  bodies  have  equal 
masses;  and  this  fact  having  been  established,  the  most  convenient 
mode  of  comparing  masses  is  by  weighing  them.   A  pound  of  iron  has 
the  same  mass  as  a  pound  of  brass  or  of  any  other  substance.     A 
pound  of  any  kind  of  matter  tends  to  the  earth  with  different  forces 
at  different  places.     The  weight  of  a  pound  of  matter  is  therefore 
not  a  definite  standard  of  force.    But  the  pound  of  matter  itself  is  a 
perfectly  definite  standard  of  mass.     If  we  weigh  one  and  the  same 
portion  of  matter  in  different  states;  for  instance  water  in  the  states 
of  ice,  snow,  liquid  water,  or  steam;  or  compare  the  weight  of  a 
chemical  compound  with  the  weights  of  its  components;  we  find  an 
exact  equality;  hence  it  has  been  stated  that  the  mass  of  a  body  is  a 
measure  of  the  quantity  of  matter  which  it  contains;  but  though 
this  statement  expresses  a  simple  fact  when  applied  to  the  compari- 
son of  different  quantities  of  one  and  the  same  substance,  it  expresses 
no  known  fact  of  nature  when  applied  to  the  comparison  of  different 
substances.     A  pound  of  iron  and  a  pound  of  lead  tend  to  the  earth 
with  equal  forces;  and  if  equal  forces  are  applied  to  them  both  their 
velocities  are  equally  affected.     We  may  if  we  please  agree  to  mea- 
sure "quantity  of  matter"  by  these  tests;  but  we  must  beware  of 
assuming  that  two  things  which  are  essentially  different  in  kind  can 
be  equal  in  themselves. 

82.  Third  Law  of  Motion.     Action  and  Reaction. — Forces  always 
occur  in  pairs,  every  exertion  of  force  being  a  mutual  action  between 
two  bodies.     Whenever  a  body  is  acted  on  by  a  force,  the  body 
from  which  this  force  proceeds  is  acted  on  by  an  equal  and  opposite 
force.     The  earth  attracts   the  moon,  and   the  moon  attracts  the 
earth.     A  magnet  attracts  iron  and  is  attracted  by  iron.     When  two 


46  FIRST  PRINCIPLES   OF   KINETICS. 

boats  are  floating  freely,  a  rope  attached  to  one  and  hauled  in  by 
a  person  in  the  other,  makes  each  boat  move  towards  the  other. 
Every  exertion  of  force  generates  equal  and  opposite  momenta  in 
the  two  bodies  affected  by  it,  since  these  two  bodies  are  acted  on  by 
equal  forces  for  equal  times. 

If  the  forces  exerted  by  one  body  upon  the  other  are  equivalent 
to  a  single  force,  the  forces  of  reaction  will  also  be  equivalent  to  a 
single  force,  and  these  two  equal  and  opposite  resultants  will  have 
the  same  line  of  action.  We  have  seen  in  §.29  that  the  general 
resultant  of  any  set  of  forces  applied  to  a  body  is  a  wrench;  that  is 
to  say  it  consists  of  a  force  with  a  definite  line  of  action  (called  the 
axis),  accompanied  by  a  couple  in  a  perpendicular  plane.  The  reac- 
tion upon  the  body  which  exerts  these  forces  will  always  be  an  equal 
and  opposite  wrench;  the  two  wrenches  having  the  same  axis,  equal 
and  opposite  forces  along  this  axis,  and  equal  and  opposite  couples 
in  the  perpendicular  plane. 

83.  Motion  of  Centre  of  Gravity  Unaffected. — A  consequence  of  the 
equality  of  the  mutual  forces  between  two  bodies  is,  that  these 
forces  produce  no  movement  of  the  common  centre  of  gravity  of  the 
two  bodies.     For  if  A  be  the  centre  of  gravity  of  a  mass  ml}  and  B 
the  centre  of  gravity  of  a  mass  m2,  their  common  centre  of  gravity 
C  will  divide  AB  inversely  as   the  masses.     Let  the  masses   be 
originally  at  rest,  and  let  them  be  acted  on  only  by  their  mutual 
attraction  or  replusion.     The  distances   through   which   they  are 
moved  by  these  equal  forces  will  be  inversely  as  the  masses,  that  is, 
will  be  directly  as  AC  and  BC;  hence  if  A'  B'  are  their  new  positions 
after  any  time,  we  have 

AC  _  AA'  _  AC  ±  AA'  _  A/C 

BC  "*  BB'  ~  BC  ±  BB'  ~  B'C' 

The  line  A'B'  is  therefore  divided  at  C  in  the  same  ratio  in  which 
the  line  AB  was  divided;  hence  C  is  still  the  centre  of  gravity. 

84.  Velocity  of  Centre  of  Gravity. — If  any  number  of  masses  are 
moving  with  any  velocities,  and  in  any  directions,  but  so  that  each 
of  them  moves  uniformly  in  a  straight  line,  their  common  centre 
of  gravity  will  move  uniformly  in  a  straight  line. 

To  prove  this,  we  shall  consider  their  component  velocities  in  any 
one  direction, 

let  these  component  velocities  be        u^        u2        u3        &c., 
the  masses  being  TOI        m^        m^        &c., 

and  the  distances  of  the  bodies  (strictly  speaking  the  distances  of 


C.G.S.   SYSTEM   OF  UNITS.  47 

their  respective  centres  of  gravity)  from  a  fixed  plane  to  which  the 
given  direction  is  normal,  be         x-^         x2         xz         &c. 

The  formula  for  the  distance  of  their  common  centre  of  gravity 
from  this  plane  is 

-  _  ml  xl  +  m,  y2  +  &c.  .j. 

nil  +  «*2  +  &c. 

In  the  time  t,  xl  is  increased  by  the  amount  uj,  x2  by  u2t,  and  so  on; 
hence  the  numerator  of  the  above  expression  is  increased  by 

OT!  M!  t  +  m^  «2  t  +  £c., 

and  the  value  of  x  is  increased  in  each  unit  of  time  by 

Ma  +  &c.  .«,. 


m.i  +  mi  +  &c. 

which  is  therefore  the  component  velocity  of  the  centre  of  gravity 
in  the  given  direction.  As  this  expression  contains  only  given 
constant  quantities,  its  value  is  constant;  and  as  this  reasoning 
applies  to  all  directions,  the  velocity  of  the  centre  of  gravity  must 
itself  be  constant  both  in  magnitude  and  direction. 

We  may  remark  that  the  above  formula  (2)  correctly  expresses 
the  component  velocity  of  the  centre  of  gravity  at  the  instant  con- 
sidered, even  when  ul}  u2,  &c.,  are  not  constant. 

85.  Centre  of  Mass.  —  The  point  which  we  have  thus  far  been 
speaking  of  under  the  name  of  "  centre  of  gravity,"  is  more  appro- 
priately called   the  "centre  of  mass,"  a   name  which   is   at   once 
suggested  by  formula  (1)  §  84.     When  gravity  acts  in  parallel  lines 
upon  all  the  particles  of  a  body,  the  resultant  force  of  gravity  upon 
the  body  is  a  single  force  passing  through  this  point;  but  this  is  no 
longer  the  case  when  the  forces  of  gravity  upon  the  different  parts 
of  tlie  body  (or  system  of  bodies)  are  not  parallel. 

86.  Units   of   Measurement.  —  It  is  a  matter  of  importance,   in 
scientific  calculations,  to  express  the  various  magnitudes  with  which 
we  have  to  deal  in  terms  of  units  which  have  a  simple  relation  to 
each  other.     The  British  weights  and  measures  are  completely  at 
fault  in  this  respect,  for  the  following  reasons:  — 

1.  They  are  not  a   decimal  system;    and    the  reduction  of  a 
measurement  (say)  from  inches  and  decimals  of  an  inch  to  feet  and 
decimals  of  a  foot,  cannot  be  effected  by  inspection. 

2.  It  is  still  more  troublesome  to  reduce  gallons  to  cubic  feet  or  inches. 

3.  The  weight  (properly  the  mass)  of  a  cubic  foot  of  a  substance 
in  Ibs.,  cannot  be  written  down  by  inspection,  when  the  specific 
gravity  of  the  substance  (as  compared  with  water)  is  givei}. 


48  FIRST   PRINCIPLES   OF   KINETICS. 

87.  The  C.G.S.  System. — A  committee  of  the  British  Association, 
specially  appointed  to  recommend  a  system  of  units  for  general 
adoption  in  scientific  calculation,  have  recommended  that  the 
centimetre  be  adopted  as  the  unit  of  length,  the  gramme  as  the  unit 
of  mass,  and  the  second  as  the  unit  of  time.  We  shall  first  give  the 
rough  and  afterwards  the  more  exact  definitions  of  these  quantities. 

The  centimetre  is  approximately  ^  of  the  distance  of  either 
pole  of  the  earth  from  the  equator;  that  is  to  say  1  followed  by  9 
zeros  expresses  this  distance  in  centimetres. 

The  gramme  is  approximately  the  mass  of  a  cubic  centimetre 
of  cold  water.  Hence  the  same  number  which  expresses  the  speci- 
fic gravity  of  a  substance  referred  to  water,  expresses  also  the  mass 
of  a  cubic  centimetre  of  the  substance,  in  grammes. 

The  second  is  24  x  *Q  x  60  of  a  mean  solar  day. 

More  accurately,  the  centimetre  is  defined  as  one  hundredth  part 
of  the  length,  at  the  temperature  0°  Centigrade,  of  a  certain  stand- 
ard bar,  preserved  in  Paris,  carefully  executed  copies  of  which 
are  preserved  in  several  other  places;  and  the  gramme  is  defined  as 
one  thousandth  part  of  the  mass  of  a  certain  standard  which  is 
preserved  at  Paris,  and  of  which  also  there  are  numerous  copies 
preserved  elsewhere. 

For  brevity  of  reference,  the  committee  have  recommended  that 
the  system  of  units  based  on  the  Centimetre,  Gramme,  and  Second, 
be  called  the  C.G.S.  system. 

The  unit  of  area  in  this  system  is  the  square  centimetre. 

The  unit  of  volume  is  the  cubic  centimetre. 

The  unit  of  velocity  is  a  velocity  of  a  centimetre  per  second. 

The  unit  of  momentum  is  the  momentum  of  a  gramme  moving 
with  a  velocity  of  a  centimetre  per  second. 

The  unit  force  is  that  force  which  generates  this  momentum  in 
one  second.  It  is  therefore  that  force  which,  acting  on  a  gramme 
for  one  second,  generates  a  velocity  of  a  centimetre  per  second. 
This  force  is  called  the  dyne,  an  abbreviated  derivative  from  the 
Greek  Sdvafiic  (force). 

The  unit  of  work  is  the  work  done  by  a  force  of  a  dyne  working 
through  a  distance  of  a  centimetre.  It  might  be  called  the  dyne- 
centimetre,  but  a  shorter  name  has  been  provided  and  it  is  called 
the  erg,  from  the  Greek  tp-yov  (work). 


CHAPTER    VII. 


LAWS   OF   FALLING  BODIES. 


88.  Effect  of  the  Resistance  of  the  Air.— In  air,  bodies  fall  with 
unequal  velocities;  a  sovereign  or  a  ball  of  lead  falls  rapidly,  a  piece 
of  down  or  thin  paper  slowly.  It  was  formerly  thought  that  this 
difference  was  inherent  in  the  nature  of  the  materials;  but  it  is 
easy  to  show  that  this  is  not  the  case,  for  if  we  compress  a  mass 
of  down  or  a  piece  of  paper  by  rolling  it  into  a  ball,  and  compare  it 
with  a  piece  of  gold-leaf,  we  shall  find  that  the  latter  body  falls 
more  slowly  than  the  former.  The  inequality  of  the  velocities 
which  we  observe  is  due  to  the  resistance  of  the  air,  which  increases 
with  the  extent  of  surface  exposed  by  the  body. 

It  was  Galileo  who  first  discovered  the  cause  of  the  unequal 
rapidity  of  fall  of  different  bodies.  To  put  the  matter  to  the  test, 
he  prepared  small  balls  of  different  substances,  and  let  them  fall  at 
the  same  time  from  the  top  of  the  tower  of  Pisa;  they  struck  the 
ground  almost  at  the  same  instant.  On  changing  their  forms,  so  as 
to  give  them  very  different  extents  of  surface,  he  observed  that  they 
fell  with  very  unequal  velocities.  He  was  thus  led  to  the  conclusion 
that  gravity  acts  on  all  substances  with  the  same  intensity,  and  that 
in  a  vacuum  all  bodies  would  fall  with  the  same  velocity. 

This  last  proposition  could  not  be  put  to  the  test  of  experiment 
in  the  time  of  Galileo,  the  air-pump  not  having  yet  been  invented. 
The  experiment  was  performed  by  Newton,  and  is  now  well  known 
as  the  "  guinea  and  feather  "  experiment.  For  this  purpose  a  tube 
from  a  yard  and  a  half  to  two  yards  long  is  used,  which  can  be 
exhausted  of  air,  and  which  contains  bodies  of  various  densities,  such 
as  a  coin,  pieces  of  paper,  and  feathers.  When  the  tube  is  full  of 
air  and  is  inverted,  these  different  bodies  are  seen  to  fall  with  very 
unequal  velocities;  but  if  the  experiment  is  repeated  after  the  tube 
4 


50  LAWS   OF   FALLING  BODIES. 

has  been  exhausted  of  air,  no  difference  can  be  perceived  between 
the  times  of  their  descent. 

89.  Mass  and  Gravitation  Proportional.— This  experiment  proves 
that  bodies  which  have  equal  weights  are  equal  in  mass.     For  equal 
masses  are  defined  to  be  those  which,  when  acted  on  by  equal  forces, 
receive  equal  accelerations;  and  the  forces,  in  this  experiment,  are 
the  weights  of  the  falling  bodies. 

Newton  tested  this  point  still  more  severely  by  experiments  with 
pendulums  (Principia,  book  III.  prop.  vi.).  He  procured  two 
round  wooden  boxes  of  the  same  size  and  weight,  and  suspended 
them  by  threads  eleven  feet  long.  One  of  them  he  filled  with  wood, 
and  he  placed  very  accurately  in  the  centre  of  oscillation  of  the 
other  the  same  weight  of  gold.  The  boxes  hung  side  by  side,  and, 
when  set  swinging  in  equal  oscillations,  went  and  returned  together 
for  a  very  long  time.  Here  the  forces  concerned  in  producing  and 
checking  the  motion,  namely,  the  force  of  gravity  and  the  resistance 
of  the  air,  were  the  same  for  the  two  pendulums,  and  as  the  move- 
ments produced  were  the  same,  it  follows  that  the  masses  were 
equal.  Newton  remarks  that  a  difference  of  mass  amounting  to  a 
thousandth  part  of  the  whole  could  not  have  escaped  detection.  He 
experimented  in  the  same  way  with  silver,  lead,  glass,  sand,  salt, 
water,  and  wheat,  and  with  the  same  result.  He  therefore  infers 
that  universally  bodies  of  equal  mass  gravitate  equally  towards  the 
earth  at  the  same  place.  He  further  extends  the  same  law  to  gravi- 
tation generally,  and  establishes  the  conclusion  that  the  mutual 
gravitating  force  between  any  two  bodies  depends  only  on  their 
masses  and  distances,  and  is  independent  of  their  materials. 

The  time  of  revolution  of  the  moon  round  the  earth,  considered  in 
conjunction  with  her  distance  from  the  earth,  shows  that  the  relation 
between  mass  and  gravitation  is  the  same  for  the  material  of  which 
the  moon  is  composed  as  for  terrestrial  matter;  and  the  same  con- 
clusion is  proved  for  the  planets  by  the  relation  which  exists  between 
their  distances  from  the  sun  and  their  times  of  revolution  in  their 
orbits. 

90.  Uniform  Acceleration. — The  fall  of  a  heavy  body  furnishes  an 
illustration  of  the  second   law  of  motion,  which  asserts  that  the 
change  of  momentum  in  a  body  in  a  given  time  is  a  measure  of  the 
force  which  acts  on  the  body.     It  follows  from  this  law  that  if  the 
same  force  continues  to  act  upon  a  body  the  changes  of  momentum 
in  successive  equal  intervals  of  time  will  be  equal.     When  a  heavy 


UNIFORM   ACCELERATION.  51 

body  originally  at  rest  is  allowed  to  fall,  it  is  acted  on  during  the 
time  of  its  descent  by  its  own  weight  and  by  no  other  force,  if  we 
neglect  the  resistance  of  the  air.  As  its  own  weight  is  a  constant 
force,  the  body  receives  equal  changes  of  momentum,  and  therefore 
of  velocity,  in  equal  intervals  of  time.  Let  g  denote  its  velocity 
in  centimetres  per  second,  at  the  end  of  the  first  second.  Then  at 
the  ead  of  the  next  second  its  velocity  will  be  g  +  g,  that  is  2g;  at 
the  end  of  the  next  it  will  be  2g+g,  that  is  3g,  and  so  on,  the  gain 
of  velocity  in  each  second  being  equal  to  the  velocity  generated  in 
the  first  second.  At  the  end  of  t  seconds  the  velocity  will  therefore 
be  tg.  Such  motion  as  this  is  said  to  be  uniformly  accelerated,  and 
the  constant  quantity  g  is  the  measure  of  the  acceleration.  Accelera- 
tion is  defined  as  the  gain  of  velocity  per  unit  of  time. 

91.  Weight  of  a  Gramme  in  Dynes.     Value  of  g. — Let  m  denote 
the  mass  of  the  falling  body  in  grammes.     Then  the  change  of 
momentum  in  each  second  is  mg,  which  is  therefore  the  measure  of 
the  force  acting  on  the  body.     The  weight  of  a  body  of  m  grammes 
is  therefore  mg  dynes,  and  the  weight  of  1  gramme  is  g  dynes.     The 
value  of  g  varies  from  9781  at  the  equator  to  983'1  at  the  poles; 
and  981  may  be  adopted  as  its  average  value  in  temperate  latitudes. 
Its  value  at  any  part  of  the  earth's  surface  is  approximately  given 

by  the  formula 

g  -  980-6056  -  2'5028  cos  2X  -  -000.003A, 

in  which  A.  denotes  the  latitude,  and  h  the  height  (in  centimetres) 
above  sea-level.1 

In  §  79  we  distinguished  between  the  intensity  and  the  amount 
of  a  force.  The  amount  of  the  force  of  gravity  upon  a  mass  of  m 
grammes  is  mg  dynes.  The  intensity  of  this  force  is  g  dynes  per 
gramme.  The  intensity  of  a  force,  in  dynes  per  gramme  of  the  body 
acted  on,  is  always  equal  to  the  change  of  velocity  which  the  force 
produces  per  second,  this  change  being  expressed  in  centimetres  per 
second.  In  other  words  the  intensity  of  a  force  is  equal  to  the 
acceleration  which  it  produces.  The  best  designation  for  g  is  the 
intensity  of  gravity. 

92.  Distance  fallen  in  a  Given  Time. — The  distance  described  in  a 
given  time  by  a  body  moving  with  uniform  velocity  is  calculated 
by  multiplying  the  velocity  by  the  time;  just  as  the  area  of  a  rect- 
angle is  calculated  by  multiplying  its  length  by  its  breadth.     Hence 
if  we  draw  a  line  such  that  its  ordinates  AA',  BB',  &c.,  represent  the 

1  For  the  method  of  determination  see  §  120. 


59  LAWS   OF   FALLING   BODIES. 

velocities  with  which  a  body  is  moving  at  the  times  represented  by 
OA,  OB  (time  being  reckoned  from  the  beginning  of  the  motion),  it 
m,  can  be  shown  that  the  whole  distance 
described  is  represented  by  the  area 
OB'B  bounded  by  the  curve,  the  last 
ordinate,  and  the  base  line.  In  fact  this 
area  can  be  divided  into  narrow  strips 
(one  of  which  is  shown  at  AA',  Fig.  37) 


Fig- 37.  each  of  which  may  practically  be  re- 

garded as  a  rectangle,  whose  height  represents  the  velocity  with 
which  the  body  is  moving  during  the  very  small  interval  of  time 
represented  by  its  base,  and  whose  area  therefore  represents  the 
distance  described  in  this  time. 

This  would  be  true  for  the  distance  described  by  a  body  moving 
from  rest  with  any  law  of  velocity.  In  the  case  of  falling  bodies 
the  law  is  that  the  velocity  is  simply  proportional  to  the  time;  hence 
the  ordinates  AA',  BB',  &c.,  must  be  directly  as  the  abscissae  OA, 
OB;  this  proves  that  the  line  OA'  B'  must  be  straight;  and  the  figure 
OB'  B  is  therefore  a  triangle.  Its  area  will  be  half  the  product  of 
OB  and  BB'.  But  OB  represents  the  time  t  occupied  by  the  motion, 
and  BB'  the  velocity  gt  at  the  end  of  this  time.  The  area  of  the 
triangle  therefore  represents  half  the  product  of  t  and  gt,  that  is, 
represents  \gtz,  which  is  accordingly  the  distance  described  in  the 
time  t.  Denoting  this  distance  by  s,  and  the  velocity  at  the  end  of 
time  t  by  v,  we  have  thus  the  two  formulas 

v  =  yt,  (1) 

«  =  to*,  (2) 

from  which  we  easily  deduce 

gs  =  {A  (3) 

93.  Work  spent  in  Producing  Motion. — We  may  remark,  in  pass- 
ing, that  the  third  of  these  formulae  enables  us  to  calculate  the  work 
required  to  produce  a  given  motion  in  a  given  mass.     When  a  body 
whose  mass  is  1  gramme  falls  through  a  distance  s,  the  force  which 
acts  upon  it  is  its  own  weight,  which  is  g  dynes,  and  the  work  done 
upon  it  is  ga  ergs.     Formula  (3)  shows  that  this  is  the  same  as  ±vz 
ergs.     For  a  mass  of  m  grammes  falling  through  a  distance  8,  the 
work  is  %mvz  ergs.     The  work  required  to  produce  a  velocity  v  (cen- 
timetres per  second)  in  a  body  of  mass  m  (grammes)  originally  at 
rest  is  %mvz  (ergs). 

94.  Body  thrown  Upwards. — When  a  heavy  body  is  projected  ver- 


WORK   IN   PRODUCING   MOTION.  53 

tically  upwards,  the  formulae  (1)  (2)  (3)  of  §  92  will  still  apply  to 
its  motion,  with  the  following  interpretations: — 

v  denotes  the  velocity  of  projection. 

t  denotes  the  whole  time  occupied  in  the  ascent. 

s  denotes  the  height  to  which  the  body  will  ascend. 
When  the  body  has  reached  the  highest  point,  it  will  fall  back,  and 
its  velocity  at  any  point  through  which  it  passes  twice  will  be  the 
same  in  going  up  as  in  coming  down. 

95.  Resistance  of  the  Air. — The  foregoing  results  are  rigorously 
applicable  to  motion  in  vacuo,  and  are  sensibly  correct  for  motion 
in  air  as  long  as  the  resistance  of  the  air  is  insignificant  in  compari- 
son with  the  force  of  gravity.     The  force  of  gravity  upon  a  body  is 
the  same  at  all  velocities;  but  the  resistance  of  the  air  increases  with 
the  velocity,  and  increases  more  and  more  rapidly  as  the  velocity 
becomes  greater;  so  that  while  at  very  slow  velocities  an  increase  of 
1  per  cent,  in  velocity  would  give  an  increase  of  1  per  cent,  in  the 
resistance,  at  a  higher  velocity  it  would  give  an  increase  of  2  per 
cent.,  and  at  the  velocity  of  a  cannon-ball  an  increase  of  3  per  cent.1 
The   formulae  are   therefore  sensibly  in   error   for   high  velocities. 
They  are  also  in  error  for  bodies  which,  like  feathers  or  gold-leaf, 
have  a  large  surface  in  proportion  to  their  weight. 

96.  Projectiles. — If,  instead  of  being  simply  let  fall,  a  body  is  pro- 
jected in  any  direction,  its  motion  will  be  compounded  of  the  motion 
of  a  falling  body  and  a  uniform  motion  in 

the  direction  of  projection.  Thus  if  OP 
(Fig.  38)  is  the  direction  of  projection,  and 
OQ  the  vertical  through  the  point  of  pro- 
jection, the  body  would  move  along  OP 
keeping  its  original  -velocity  unchanged,  if 
it  were  not  disturbed  by  gravity.  To  find  Q  Fig.  38. 

where  the  body  will  be  at  any  time  t,  we  must 

lay  off  a  length  OP  equal  to  V£,  V  denoting  the  velocity  of  projec- 
tion, and  must  then  draw  from  P  the  vertical  line  PR  downwards 
equal  to  kgt2,  which  is  the  distance  that  the  body  would  have  fallen 
in  the  time  if  simply  dropped.  The  point  R  thus  determined,  will 
be  the  actual  position  of  the  body.  The  velocity  of  the  body  at 
any  time  will  in  like  manner  be  found  by  compounding  the  initial 

1  This  is  only  another  way  of  saying  that  the  resistance  varies  approximately  as  tho 
velocity  when  very  small,  and  approximately  as  the  cube  of  the  velocity  for  velocities  like 
that  of  a  cannon-ball. 


54  LAWS   OF   FALLING   BODIES. 

velocity  with  the  velocity  which  a  falling  body  would  have  acquired 
in  the  time. 

The  path  of  the  body  will  be  a  curve,  as  represented  in  the 
figure,  OP  being  a  tangent  to  it  at  0,  and  its  concavity  being  down- 
wards. The  equations  above  given,  namely 

show  that  PR  varies  as  the  square  of  OP,  and  hence  that  the  path 
(or  trajectory  as  it  is  technically  called)  is  a  parabola,  whose  axis  is 
vertical. 

97.  Time  of  Flight,  and  Range.— If  the  body  is  projected  from  a 
point  at  the  surface  of  the  ground  (supposed  level)  we  can  calculate 
the  time  of  flight  and  the  range  in  the  following  way. 

Let  a  be  the  angle  which  the  direction  of  projection  makes  with  the 
horizontal.  Then  the  velocity  of  projection  can  be  resolved  into 
two  components,  V  cos  a  and  V  sin  a,  the  former  being  horizontal, 
and  the  latter  vertically  upward.  The  horizontal  component  of  the 
velocity  of  the  body  is  unaffected  by  gravity  and  remains  constant. 
The  vertical  velocity  after  time  t  will  be  compounded  of  V  sin  a  up- 
wards and  gt  downwards.  It  will  therefore  be  an  upward  velocity 
V  sin  a  —  gt,  or  a  downward  velocity  gt  —  V  sin  a.  At  the  highest 
point  of  its  path,  the  body  will  be  moving  horizontally  and  the  ver- 
tical component  of  its  velocity  will  be  zero;  that  is,  we  shall  have 


This  is  the  time  of  attaining  the  highest  point;   and  the  time  of 
flight  will  be  double  of  this,  that  is,  will  be  2V  sin  tt. 

As  the  horizontal  component  of  the  velocity  has  the  constant 
value  V  cos  a,  the  horizontal  displacement  in  any  time  t  is  V  cos  a 
multiplied  by  t  The  range  is  therefore 

2V2  sin  a  cos  a       V8  sin  2a 
ff  </ 

The  range  (for  a  given  velocity  of  projection)  will  therefore  be 
greatest  when  sin  2a  is  greatest,  that  is  when  2o  =  90°  and  a=45°. 

We  shall  now  describe  two  forms  of  apparatus  for  illustrating  the 
laws  of  falling  bodies. 

98.  Morin's  Apparatus. — Morin's  apparatus  consists  of  a  wooden 
cylinder  covered  with  paper,  which  can  be  set  in  unifo'rm  rotation 
about  its  axis  by  the  fall  of  a  heavy  weight.  The  cord  which  sup- 


PROJECTILES 


55 


ports  the  weight  is  wound  upon  a  drum,  furnished  with  a  toothed 
wheel  which  works  on  one  side  with  an  endless  screw  on  the  axis 
of  the  cylinder,  and  on  the  other  drives  an  axis  carrying  fans  which 
serve  to  regulate  the  motion. 

In  front  of  the  turning  cylinder  is  a  cylindro-conical  weight  of 
cast-iron  carrying  a  pen- 
cil whose  point  presses 
against  the  paper,  and 
having  ears  which  slide 
on  vertical  threads,  serv- 
ing to  guide  it  in  its  fall. 
By  pressing  a  lever,  the 
weight  can  be  made  to 
fall  at  a  chosen  moment. 
The  proper  time  for  this 
is  when  the  motion  of 
the  cylinder  has  become 
sensibly  uniform.  It  fol- 
lows from  this  arrange- 
ment that  during  its 
vertical  motion  the  pencil 
will  meet  in  succession 
the  different  generating 
lines1  of  the  revolving 
cylinder,  and  will  conse- 
quently describe  on  its 
surface  a  certain  curve,  _ 

from  the  study  o£  which  !  ^^^g^ 
we  shall  be  able  to  gather 
the  law  of  the  fall  of  the 
body  which  has  traced 
it.  With  this  view,  we 
describe  (by  turning  the 
cylinder  while  the  pencil 
is  stationary)  a  circle  passing  through  the  commencement  of  the 
curve,  and  also  draw  a  vertical  line  through  this  point.  We  cut 
the  paper  along  this  latter  line  and  develop  it  (that  is,  flatten 

1  A  cylindric  surface  could  be  swept  out  or  "generated"  by  a  straight  line  moving 
round  the  axis  and  remaining  always  parallel  to  it.  The  successive  positions  of  this 
generating  line  are  called  the  "  generating  lines  of  the  cylinder." 


Fig.  39. — M  oriii's  Apparatus. 


56 


LAWS   OF   FALLING  BODIES. 


tances  1,  2,  3,  4,  5. 


it  out  into  a  plane).     It  then  presents  the  appearance  shown  in 

If  we  take  on  the  horizontal  line  equal  distances  at  1,  2,  3,  4,  5 
,  and  draw  perpendiculars  at  their  extremities  to  meet  the 
curve,'  it 'is  evident  that  the  points  thus  found  are  those  which  were 
traced  by  the  pencil  when  the  cylinder  had  turned  through  the  dis- 
The  corresponding  verticals  represent 
the  spaces  traversed  in  the  times  1,  2,  3, 
4,  5.    ...    Now  we  find,  as  the  figure 
shows,  that  these  spaces  are  represented 
by  the  numbers  1,  4,  9,  16,  25    .     .     .     , 
thus  verifying  the  principle  that  the  spaces 
described  are  proportional  to  the  squares 
of  the  times  employed  in  their  description. 
We  may  remark  that  the  proportionality 
of  the  vertical  lines  to  the  squares  of  the 
horizontal  lines  shows  that  the  curve  is  a 
parabola.     The  parabolic  trace  is  thus  the 
consequence  of  the  law  of  fall,  and  from 
the    fact    of    the    trace    being    parabolic 
we  can  infer  the  proportionality  of  the 
spaces  to  the  squares  of  the  times. 
The  law  of  velocities  might  also  be  verified  separately  by  Morin's 
apparatus;  we  shall  not  describe  the  method  which  it  would  be 
necessary  to  employ,  but  shall  content  ourselves  with  remarking 
that  the  law  of  velocities  is  a  logical  consequence  of   the  law  of 
spaces.1 

99.  Atwood's  Machine. — Atwood's  machine,  which  affords  great 
facilities  for  illustrating  the  effects  of  force  in  producing  motion, 
consists  essentially  of  a  very  freely  moving  pulley  over  which  a  fine 
cord  passes,  from  the  ends  of  which  two  equal  weights  can  be  sus- 
pended. A  small  additional  weight  of  flat  and  elongated  form  is 
laid  upon  one  of  them,  which  is  thus  caused  to  descend  with  uni- 
form acceleration,  and  means  are  provided  for  suddenly  removing 

1  Consider,  in  fact,  the  space  traversed  in  any  time  t ;  this  space  is  given  by  the  formula 
s  =  Kt-;  during  the  time  t  +  0  the  space  traversed  will  be  K(t  +  0)*  =  Kt*  +  2KW  +  ~KO\ 
whence  it  follows  that  the  space  traversed  during  the  time  0  after  the  time  t  is  2K<  6  + 
K02.  The  average  velocity  during  this  time  6  is  obtained  by  dividing  the  space  by  6, 
and  is  2K«  +  K0,  which,  by  making  6  very  small,  can  be  made  to  agree  as  accurately  as 
we  please  with  the  value  2Kt.  This  limiting  value  2Kt  must  therefore  be  the  velocity  at 
the  end  of  time  t. — Z>. 


Fig.  40.— Parabolic  Trace 


ATWOODS   MACHINE. 


57 


this  additional  weight  at  any  point  of  the  descent,  so  as  to  allow  the 
motion  to  continue  from 
this  point  onward  with 
uniform  velocity. 

The  machine  is  re- 
presented in  Fig.  41. 
The  pulley  over  which 
the  string  passes  is  the 
largest  of  the  wheels 
shown  at  the  top  of  the 
apparatus.  In  order  to 
give  it  greater  freedom 
of  movement,  the  ends 
of  its  axis  are  made 
to  rest,  not  on  fixed 
supports,  but  on  the 
circumferences  of  four 
wheels  (two  at  each 
end  of  the  axis)  called 
friction-wheels,  because 
their  office  is  to  dim- 
inish friction.  Two 
small  equal  weights  are 
shown,  suspended  from 
this  pulley  by  a  string 
passing  over  it.  One  of 
them  P'  is  represented 
as  near  the  bottom  of 
the  supporting  pillar, 
and  the  other  P  as  near 
the  top.  The  latter  is 
resting  upon  a  small 
platform,  which  can  be 
suddenly  dropped  when 
it  is  desired  that  the 
motion  shall  commence. 
A  little  lower  down  and 
vertically  beneath  the 
platform,  is  seen  a  ring,  Fl's-  «-Atwo«r.  Machine. 

large  enough  to  let  the  weight  pass  through  it  without  danger  of 


58  LAWS   OF   FALLING   BODIES. 

contact.  This  ring  can  be  shifted  up  or  down,  and  clamped  at  any 
height  by  a  screw;  it  is  represented  on  a  larger  scale  in  the  margin. 
At  a  considerable  distance  beneath  the  ring,  is  seen  the  stop,  which 
is  also  represented  in  the  margin,  and  can  like  the  ring  be  clamped 
at  any  height.  The  office  of  the  ring  is  to  intercept  the  additional 
weight,  and  the  office  of  the  stop  is  to  arrest  the  descent.  The  up- 
right to  which  they  are  both  clamped  is  marked  with  a  scale  of  equal 
parts,  to  show  the  distances  moved  over.  A  clock  with  a  pendulum 
beating  seconds,  is  provided  for  measuring  the  time;  and  there  is  an 
arrangement  by  which  the  movable  platform  can  be  dropped  by  the 
action  of  the  clock  precisely  at  one  of  the  ticks.  To  measure  the 
distance  fallen  in  one  or  more  seconds,  the  ring  is  removed,  and  the 
stop  is  placed  by  trial  at  such  heights  that  the  descending  weight 
strikes  it  precisely  at  another  tick.  To  measure  the  velocity 
acquired  in  one  or  more  seconds,  the  ring  must  be  fixed  at  such  a 
height  as  to  intercept  the  additional  weight  at  one  of  the  ticks,  and 
the  stop  must  be  placed  so  as  to  be  struck  by  the  descending  weight 
at  another  tick. 

100.  Theory  of  Atwood's  Machine.  —  If  M  denote  each  of  the  two 
equal  masses,  in  grammes,  and  m  the  additional  mass,  the  whole 
moving  mass  (neglecting  the  mass  of  the  pulley  and  string)  is 
2M  +  m,  but  the  moving  force  is  only  the  weight  of  m.  The  accel- 
eration produced,  instead  of  being  g,  is  accordingly  only  -^  —  g. 
In  order  to  allow  for  the  inertia  of  the  pulley  and  string,  a  con- 
stant quantity  must  be  added  to  the  denominator  in  the  above  for- 
mula, and  the  value  of  this  constant  can  be  determined  by  observ- 
ing the  movements  obtained  with  different  values  of  M  and  m. 
Denoting  it  by  C,  we  have 


as  the  expression  for  the  acceleration.  As  m  is  usually  small  in 
comparison  with  M,  the  acceleration  is  very  small  in  comparison  with 
that  of  a  freely  falling  body,  and  is  brought  within  the  limits  of 
convenient  observation.  Denoting  the  acceleration  by  a,  and  using 
v  and  s,  as  in  §  92,  to  denote  the  velocity  acquired  and  space 
described  in  time  t,  we  shall  have 


=  at,  (1) 

=4<  (2) 

=W,  (3) 


FORCE   IN   CIRCULAR  MOTION. 


59 


and  each  of  these  formulae  can  be  directly  verified  by  experiments 
with  the  machine. 

101.  Uniform  Motion  in  a  Circle.— 
A  body  cannot  move  in  a  curved  path 
unless  there  be  a  force  urging  it  Fi*-42- 

towards  the  concave  side  of  the  curve.  We  shall  proceed  to  in- 
vestigate the  intensity  of  this  force  when 
the  path  is  circular  and  the  velocity  uniform. 
We  shall  denote  the  velocity  by  v,  the  radius 
of  the  circle  by  r,  and  the  intensity  of  the 
force  by  /.  Let  AB  (Figs.  42, 43)  be  a  small 
portion  of  the  path,  and  BD  a  perpendicular 
upon  AD  the  tangent  at  A.  Then,  since 
the  arc  AB  is  small  in  comparison  with 
the  whole  circumference,  it  is  sensibly  equal 
to  AD,  and  the  body  would  have  been  found 
at  D  instead  of  at  B  if  no  force  had  acted 

upon  it  since  leaving  A.  DB  is  accordingly  the  distance  due  to  the 
force;  and  if  t  denote  the  time  from  A  to  B,  we  have 

AD  =  vt  (1) 

DB  =  i/c2.  (2) 

The  second  of  these  equations  gives 

/=  — 
and  substituting  for  t  from  the  first  equation,  this  becomes 

f  =  S§  *  (3) 

But  if  An  (Fig.  43)  be  the  diameter  at  A,  and  Bra  the  perpendicular 
upon  it  from  B,  we  have,  by  Euclid,  AD2  =  mB2— Am.mn=2r.  Am 
sensibly, -2r.DB. 

Therefore  33^=  p  and  hence  by  (3) 

/=£  («> 

Hence  the  force  necessary  for  keeping  a  body  in  a  circular  path 
without  change  of  velocity,  is  a  force  of  intensity  -  directed  towards 
the  centre  of  the  circle.  If  m  denote  the  mass  of  the  body,  the 
amount  of  the  force  will  be  ^-.  This  will  be  in  dynes,  if  m  be  in 
grammes,  r  in  centimetres,  and  v  in  centimetres  per  second. 

If  the  time  of  revolution  be  denoted  by  T,  and  TT  as  usual  denote 
the  ratio  of  circumference  to  diameter,  the  distance  mov^d  in  time 


60  LAWS   OF   FALLING   BODIES. 

T  is  2wr;  hence  v  =  ^r,  and  another  expression  for  the  intensity  of 
the  force  will  be 

f=("ffr.  M 

102.  Deflecting  Force  in  General.— In  general,  when  a  body  is 
moving  in  any  path,  and  with  velocity  either  constant  or  varying, 
the  force  acting  upon  it  at  any  instant  can  be  resolved  into  two 
components,  one  along  the  tangent  and  the  other  along  the  normal. 
The  intensity  of  the  tangential  component  is  measured  by  the  rate 
at  which  the  velocity  increases  or  diminishes,  and  the  intensity  of 
the  normal  component  is  given  by  formula  (4)  of  last  article,  if  we 
make  T  denote  the  radius  of  curvature. 

103.  Illustrations  of  Deflecting   Force. — When  a  stone  is  swung 
round  by  a  string  in  a  vertical  circle,  the  tension  of  the  string  in 
the  lowest  position  consists  of  two  parts: — 

(1)  The  weight  of  the  stone,  which  is  mg  if  m  be  the  mass  of  the 
stone. 

(2)  The  force  m  -  which  is  necessary  for  deflecting  the  stone  from 
a  horizontal  tangent  into  its  actual  path  in  the  neighbourhood  of  the 
lowest  point. 

When  the  stone  is  at  the  highest  point  of  its  path,  the  tension  of 
the  string  is  the  difference  of  these  two  forces,  that  is  to  say  it  is 

and  the  motion  is  not  possible  unless  the  velocity  at  the  highest 
point  is  sufficient  to  make  £  greater  than  g. 

The  tendency  of  the  stone  to  persevere  in  rectilinear  motion  and 
to  resist  deflection  into  a  curve,  causes  it  to  exert  a  force  upon  the 
string,  of  amount  m  v-,  and  this  is  called  centrifugal  force.  It  is 
not  a  force  acting  upon  the  stone,  but  a  force  exerted  by  the  stone 
upon  the  string.  Its  direction  is  from  the  centre  of  curvature, 
whereas  the  deflecting  force  which  acts  upon  the  stone  is  towards 
the  centre  of  curvature. 

104.  Centrifugal   Force  at  the   Equator.— Bodies  on   the  earth's 
surface  are  carried  round  in  circles  by  the  diurnal  rotation  of  the 
earth  upon  its  axis.     The  velocity  of  this  motion  at  the  equator  is 
about   46,500  centimetres   per   second,  and   the  earth's   equatorial 
radius  is  about  6*38  x  108  centimetres.     Hence  the  value  of  -  is 
found  to  be  about  3'39.     The  case  is  analogous  to  that  of  the  stone 


APPARENT   GRAVITY.  61 

at  the  highest  point  of  its  path  in  the  preceding  article,  if  instead 
of  a  string  which  can  only  exert  a  pull  we  suppose  a  stiff  rod  which 
can  exert  a  push  upon  the  stone.  The  rod  will  be  called  upon  to 
exert  a  pull  or  a  push  at  the  highest  point  according  as  -  is  greater 
or  less  than  g.  The  force  of  the  push  in  the  latter  case  will  be 


m\  g-- 


and  this  is  accordingly  the  force  with  which  the  surface  of  the  earth 
at  the  equator  pushes  a  body  lying  upon  it.  The  push,  of  course, 
is  mutual,  and  this  formula  therefore  gives  the  apparent  weight  or 
apparent  gravitating  force  of  a  body  at  the  equator,  mg  denoting  its 
true  gravitating  force  (due  to  attraction  alone).  A  body  falling  in 
vacuo  at  the  equator  has  an  acceleration  978*10  relative  to  the 
surface  of  the  earth  in  its  neighbourhood;  but  this  portion  of  the 
surface  has  itself  an  acceleration  of  3*39,  directed  towards  the  earth's 
centre,  and  therefore  in  the  same  direction  as  the  acceleration  of  the 
body.  The  absolute  acceleration  of  the  body  is  therefore  the  sum  of 
these  two,  that  is  981*49,  which  is  accordingly  the  intensity  of  true 
gravity  at  the  equator. 

The  apparent  weight  of  bodies  at  the  equator  would  be  nil  if  v- 
were  equal  to  g.  Dividing  3*39  into  981'49,  the  quotient  is  approxi- 
mately 289,  which  is  (17)2.  Hence  this  state  of  things  would  exist 
if  the  velocity  of  rotation  were  about  17  times  as  fast  as  at  present. 

Since  the  movements  and  forces  which  we  actually  observe  depend 
upon  relative  acceleration,  it  is  usual  to  understand,  by  the  value  of 
g  or  the  intensity  of  gravity  at  a  place,  the  apparent  values,  unless 
the  contrary  be  expressed.  Thus  the  value  of  g  at  the  equator  is 
usually  stated  to  be  97810. 

105.  Direction  of  Apparent  Gravity. — The  total  amount  of  centri- 
fugal force  at  different  places  on  the  earth's  surface,  varies  directly 
as  their  distance  from  the  earth's  axis;  for  this  is  the  value  of  r  in 
the  formula  (5)  of  §  101,  and  the  value  of  T  in  that  formula  is  the 
same  for  the  whole  earth.  The  direction  of  this  force,  being  per- 
pendicular to  the  earth's  axis,  is  not  vertical  except  at  the  equator; 
and  hence,  when  we  compound  it  with  the  force  of  true  gravity,  we 
obtain  a  resultant  force  of  apparent  gravity  differing  in  direction  as 
well  as  in  magnitude  from  true  gravity.  What  is  always  understood 
by  a  vertical,  is  the  direction  of  apparent  gravity;  and  a  plane  per- 
pendicular to  it  is  what  is  meant  by  a  horizontal  plane. 


CHAPTER   VIII. 


THE  PENDULUM. 


106.  The  Pendulum. — When  a  body  is  suspended  so  that  it  can  turn 
about  a  horizontal  axis  which  does  not  pass  through 
its  centre  of  gravity,  its  only  position  of  stable  equi- 
librium is  that  in  which  its  centre  of  gravity  is  in 
the  same  vertical  plane  with  the  axis  and  below  it 
(§  42).  If  the  body  be  turned  into  any  other  position, 
and  left  to  itself,  it  will  oscillate  from  one  side  to  the 
other  of  the  position  of  equilibrium,  until  the  resistance 
of  the  air  and  the  friction  of  the  axis  gradually  bring 
it  to  rest.  A  body  thus  suspended,  whatever  be  its 
form,  is  called  a  pendulum.  It  frequently  consists 
of  a  rod  which  can  turn  about  an  axis  O  (Fig.  44)  at 
its  upper  end,  and  which  carries  at  its  lower  end  a 
heavy  lens-shaped  piece  of  metal  M  called  the  bob;  this 
latter  can  be  raised  or  lowered  by  means  of  the  screw 
V.  The  applications  of  the  pendulum  are  very  impor- 
tant: it  regulates  our  clocks,  and  it  has  enabled  us  to 
measure  the  intensity  of  gravity  in  different  parts  of 
the  world;  it  is  important  then  to  know  at  least  the 
fundamental  points  in  its  theory.  For  explaining 
these,  we  shall  begin  with  the  consideration  of  an 
ideal  body  called  the  simple  pendulum. 

107.  Simple  Pendulum. — This  is  the  name  given  to  a 
pendulum  consisting  of  a  heavy  particle  M  (Fig.  45) 
attached  to  one  end  of  an  inextensible  thread  without 
weight,  the  other  end  of  the  thread  being  fixed  at  A. 
When  the  thread  is  vertical,  the  weight  of  the  particle 
K*  14. -Pendulum,  acts  in  the  direction  of  its  length,  and  there  is  equilib- 


SIMPLE   PENDULUM. 


G3 


Fig.  45.— Motion  of  Simple 
Pendulum. 


rium.  But  suppose  it  is  drawn  aside  into  another  position,  as  AM. 
In  this  case,  the  weight  MG  of  the  particle  can  be  resolved  into  two 
forces  MC  and  MH.  The  former,  acting  along  the  prolongation  of 
the  thread,  is  destroyed  by  the  resistance  of  the  thread;  the  other, 
acting  along  the  tangent  MH,  produces  the 
motion  of  the  particle.  This  effective  com- 
ponent is  evidently  so  much  the  greater  as 
the  angle  of  displacement  from  the  vertical 
position  is  greater.  The  particle  will  there- 
fore move  along  an  arc  of  a  circle  described 
from  A  as  centre,  and  the  force  which 
urges  it  forward  will  continually  diminish 
till  it  arrives  at  the  lowest  point  M'. 
At  M'  this  force  is  zero,  but,  in  virtue 
of  the  velocity  acquired,  the  particle  will 
ascend  on  the  opposite  side,  the  effective 
component  of  gravity  being  now  opposed 
to  the  direction  of  its  motion;  and,  inas- 
much as  the  magnitude  of  this  component 
goes  through  the  same  series  of  values  in  this  part  of  the  motion 
as  in  the  former  part,  but  in  reversed  order,  the  velocity  will,  in  like 
manner,  retrace  its  former  values,  and  will  become  zero  when  the 
particle  has  risen  to  a  point  M"  at  the  same  height  as  M.  It  then 
descends  again  and  performs  an  oscillation  from  M"  to  M  precisely 
similar  to  the  first,  but  in  the  reverse  direction.  It  will  thus 
continue  to  vibrate  between  the  two  points  M,  M"  (friction  being 
supposed  excluded),  for  an  indefinite  number  of  times,  all  the  vibra- 
tions being  of  equal  extent  and  performed  in  equal  periods. 

The  distance  through  which  a  simple  pendulum  travels  in  moving 
from  its  lowest  position  to  its  furthest  position  on  either  side,  is 
called  its  amplitude.  It  is  evidently  equal  to  half  the  complete  arc 
of  vibration,  and  is  commonly  expressed,  not  in  linear  measure,  but 
in  degrees  of  arc.  Its  numerical  value  is  of  course  equal  to  that  of 
the  angle  MAM',  which  it  subtends  at  the  centre  of  the  circle. 

The  complete  period  of  the  pendulum's  motion  is  the  time  which 
it  occupies  in  moving  from  M  to  M"  and  back  to  M,  or  more  generally, 
is  the  time  from  its  passing  through  any  given  position  to  its  next 
passing  through  the  same  position  in  the  same  direction. 

What  is  commonly  called  the  time  of  vibration,  or  the  time  of  a 
single  vibration,  is  the  half  of  a  complete  period,  being  the  time  of 


(54,  THE   PENDULUM. 

passing  from  one  of  the  two  extreme  positions  to  the  other.  Hence 
what  we  have  above  defined  as  a  complete  period  is  often  called  a 
double  vibration. 

When  the  amplitude  changes,  the  time  of  vibration  changes  also, 
being  greater  as  the  amplitude  is  greater;  but  the  connection  between 
the  two  elements  is  very  far  from  being  one  of  simple  proportion. 
The  change  of  time  (as  measured  by  a  ratio)  is  much  less  than  the 
change  of  amplitude,  especially  when  the  amplitude  is  small;  and 
when  the  amplitude  is  less  than  about  5°,  any  further  diminution 
of  it  has  little  or  no  sensible  effect  in  diminishing  the  time.  For 
small  vibrations,  then,  the  time  of  vibration  is  independent  of  the 
amplitude.  This  is  called  the  law  of  isochronism. 

108.  Law  of  Acceleration  for  Small  Vibrations.— Denoting  the 
length  of  a  simple  pendulum  by  I,  and  its  inclination  to  the  vertical 
at  any  moment  by  &,  we  see  from  Fig.  45  that  the  ratio  of  the  effective 
component  of  gravity  to  the  whole  force  of  gravity  is  j^-,  that 
is  sin  6;  and  when  0  is  small  this  is  sensibly  equal  to  6  itself  as 
measured  by  •  "£?  .  Let  s  denote  the  length  of  the  arc  MM'  inter- 
vening between  the  lower  end  of  the  pendulum  and  the  lowest  point 
of  its  swing,  at  any  time;  then  0  is  equal  to  -^,  and  the  intensity 
of  the  effective  force  of  gravity  when  0  is  small  is  sensibly  equal  to 
gd,  that  is  to  ^*.  Since  g  and  I  are  the 
same  in  all  positions  of  the  pendulum,  this 
effective  force  varies  as  s.  Its  direction 
is  always  towards  the  position  of  equilib- 
rium, so  that  it  accelerates  the  motion 
during  the  approach  to  this  position,  and 
retards  it  during  the  recess;  the  acceleration 
or  retardation  being  always  in  direct  pro- 
portion to  the  distance  from  the  position  of 
equilibrium.  This  species  of  motion  is  of 
extremely  common  occurrence.  It  is  illus- 
trated by  the  vibration  of  either  prong 
of  a  tuning-fork,  and  in  general  by  the 
motion  of  any  body  vibrating  in  one  plane 


\ 


Fig.  46.— Projection  of  Circular 


in  such  a  manner  as  to  yield  a  simple  musical  tone. 

109.  General  Law  for  Period.— Suppose  a  point  P  to  travel  with 
uniform  velocity  round  a  circle  (Fig.  46),  and  from  its  successive 


SIMPLE   HARMONIC   MOTION.  65 


positions  Pp  P2,  &c.,  let  perpendiculars  P^,  P2p2,  &c.,  be  drawn  to  a 
fixed  straight  line  in  the  plane  of  the  circle.  Then  while  P  travels 
once  round  the  circle,  its  projection  p  executes  a  complete  vibration. 
The  acceleration  of  P  is  always  directed  towards  the  centre  of  the 
circle,  and  is  equal  to  (Y)  r  (§  1^1)-  The  component  of  this  acceler- 
ation parallel  to  the  line  of  motion  of  p,  is  the  fraction  -  of  the  whole 
acceleration  (x  denoting  the  distance  of  p  from  the  middle  point  of 
its  path),  and  is  therefore  \-f)x-  This  is  accordingly  the  accelera- 
tion of  p,  and  as  it  is  simply  proportional  to  x  we  shall  denote  it  for 
brevity  by  px.  To  compute  the  periodic  time  T  of  a  complete 
vibration,  we  have  the  equation  /*=  (Y)*'  which  gives 

T=^.  (1) 

VM 

/   110.  Application  to  the  Pendulum.—  For  the  motion  of  a  pendulum 
in  a  small  arc,  we  have 

acceleration  =|a, 

where  s  denotes   the  displacement   in  linear   measure.     We  must 
therefore  put  M  -  j,  and  we  then  have 


which  is  the  expression  for  the  time  of  a  complete  (or  double)  vibra- 
tion. It  is  more  usual  to  understand  by  the  "  time  of  vibration  "  of 
a  pendulum  the  half  of  this,  that  is  the  time  from  one  extreme 
position  to  the  other,  and  to  denote  this  time  by  T.  In  this  sense 
we  have 


To  find  the  length  of  the  seconds'  pendulum  we  must  put  T  =  1. 
This    ives 


If  g  were  987  we  should  have  Z  =  100  centimetres  or  1  metre.  The 
actual  value  of  g  is  everywhere  a  little  less  than  this.  The  length 
of  the  seconds'  pendulum  is  therefore  everywhere  rather  less  than  a 
metre. 

111.  Simple  Harmonic  Motion.  —  Rectilinear  motion  consisting  of 
vibration   about   a   point   with   acceleration   px,  where   x   denotes 
5 


QQ  THE  PENDULUM. 

distance  from  this  point,  is  called  Simple  Harmonic  Motion,  or 
Simple  Harmonic  Vibration.  The  above  investigation  shows  that 
such  vibration  is  isochronous,  its  period  being  ^j  whatever  the 

amplitude  may  be. 

To  understand  the  reason  of  this  isochronism  we  have  only  to 
remark  that,  if  the  amplitude  be  changed,  the  velocity  at  correspond- 
ing points  (that  is,  points  whose  distances  from  the  middle  point  are 
the  same  fractions  of  the  amplitudes)  will  be  changed  in  the  same 
ratio.  For  example,  compare  two  simple  vibrations  in  which  the 
values  of  /z  are  the  same,  but  let  the  amplitude  of  one  be  double  that 
of  the  other.  Then  if  we  divide  the  paths  of  both  into  the  same 
number  of  small  equal  parts,  these  parts  will  be  twice  as  great  for 
the  one  as  for  the  other;  but  if  we  suppose  the  two  points  to  start 
simultaneously  from  their  extreme  positions,  the  one  will  constantly 
be  moving  twice  as  fast  as  the  other.  The  number  of  parts  described 
in  any  given  time  will  therefore  be  the  same  for  both. 

In  the  case  of  vibrations  which  are  not  simple,  it  is  easy  to  see 
(from  comparison  with  simple  vibration)  that  if  the  acceleration  in- 
creases in  a  greater  ratio  than  the  distance  from  the  mean  position, 
the  period  of  vibration  will  be  shortened  by  increasing  the  amplitude; 
but  if  the  acceleration  increases  in  a  less  ratio  than  the  distance,  as 
in  the  case  of  the  common  pendulum  vibrating  in  an  arc  of  moderate 
extent,  the  period  is  increased  by  increasing  the  amplitude. 

112.  Experimental  Investigation  of  the  Motion  of  Pendulums. — The 
preceding  investigation  applies  to  the  simple  pendulum;  that  is  to 
say  to  a  purely  imaginary  existence;  but  it  can  be  theoretically 
demonstrated  that  every  rigid  body  vibrating  about  a  horizontal 
axis  under  the  action  of  gravity  (friction  and  the  resistance  of  the 
air  being  neglected),  moves  in  the  same  manner  as  a  simple  pendu- 
lum of  determinate  length  called  the  equivalent  simple  pendulum. 
Hence  the  above  results  can  be  verified  by  experiments  on  actual 
pendulums. 

The  discovery  of  the  experimental  laws  of  the  motion  of  pendu- 
lums was  in  fact  long  anterior  to  the  theoretical  investigation. 
It  was  the  earliest  and  one  of  the  most  important  discoveries  of 
Galileo,  and  dates  from  the  year  1582,  when  he  was  about  twenty 
years  of  age.  It  is  related  that  on  one  occasion,  when  in  the 
cathedral  of  Pisa,  he  was  struck  with  the  regularity  of  the  oscilla- 
tions of  a  lamp  suspended  from  the  roof,  and  it  appeared  to  him 


CYCLOIDAL   PENDULUM.  67 

that  these  oscillations,  though  diminishing  in  extent,  preserved  the 
same  duration.  He  tested  the  fact  by  repeated  trials,  which  con- 
firmed him  in  the  belief  of  its  perfect  exactness.  This  law  of 
isochronism  can  be  easily  verified.  It  is  only  necessary  to  count 
the  vibrations  which  take  place  in  a  given  time  with  different 
amplitudes.  The  numbers  will  be  found  to  be  exactly  the  same. 
This  will  be  found  to  hold  good  even  when  some  of  the  vibrations 
compared  are  so  small  that  they  can  only  be  observed  with  a 
telescope. 

By  employing  balls  suspended  by  threads  of  different  lengths, 
Galileo  discovered  the  influence  of  length  on  the  time  of  vibration. 
He  ascertained  that  when  the  length  of  the  thread  increases,  the 
time  of  vibration  increases  also;  not,  however,  in  proportion  to  the 
length  simply,  but  to  its  square  root. 

113.  Cycloidal  Pendulum. — It  is  obvious  from  §  64  that  the  effective 
component  of  gravity  upon  a  particle  resting  on  a  smooth  inclined 
plane  is  proportional  to  the  sine  of  the  inclination.  The  accelera- 
tion of  a  particle  so  situated  is  in  fact  g  sin  a,  if  a  denote  the  inclina- 
tion of  the  plane.  When  a  particle  is  guided  along  a  smooth  curve 
its  acceleration  is  expressed  by  the  same  formula,  o  now  denoting  the 
inclination  of  the  curve  at  any  point  to  the  horizon.  This  inclina- 
tion varies  from  point  to  point  of  the  curve,  so  that  the  acceleration 
g  sin  o  is  no  longer  a  constant  quantity.  The  motion  of  a  common 
pendulum  corresponds  to  the  motion  of  a  particle  which  is  guided  to 
move  in  a  circular  arc;  and  if  x  denote  distance  from  the  lowest 
point,  measured  along  the  arc,  and  r  the  radius  of  the  circle  (or 

the  length  of  the  pendulum),  the  acceleration  at  any  point  is  g  sin  *• 
This  is  sensibly  proportional  to  x  so  long  as  a?  is  a  small  fraction 
of  r;  but  in  general  it  is  not  proportional  to  x,  and  hence  the  vibra- 
tions are  not  in  general  isochronous. 

To  obtain  strictly  isochronous  vibrations  we  must  substitute  for 
the  circular  arc  a  curve  which  possesses  the  property  of  having  an 
inclination  whose  sine  is  simply  proportional  to  distance  measured 
along  the  curve  from  the  lowest  point.  The  curve  which  possesses 
this  property  is  the  cycloid.  It  is  the  curve  which  is  traced  by  a 
point  in  the  circumference  of  a  circle  which  rolls  along  a  straight 
line.  The  cycloidal  pendulum  is  constructed  by  suspending  an  ivory 
ball  or  some  other  small  heavy  body  by  a  thread  between  two 
cheeks  (Fig.  47),  on  which  the  thread  winds  as  the  ball  swings  to 


08 


THE   PENDULUM. 


Fig  47.— Cycloidal  Pendulum. 


either  side.  The  cheeks  must  themselves  be  the  two  halves  of  a 
cycloid  whose  length  is  double  that  of  the  thread,  so  that  each 

cheek  has  the  same  length  as  the 
thread.  It  can  be  demonstrated1 
that  under  these  circumstances 
the  path  of  the  ball  will  be  a 
cycloid  identical  with  that  to 
which  the  cheeks  belong.  Ne- 
glecting friction  and  the  rigidity 
of  the  thread,  the  acceleration  in 
this  case  is  proportional  to  dis- 
tance measured  along  the  cycloid 
from  its  lowest  point,  and  hence 
the  time  of  vibration  will  be 
strictly  the  same  for  large  as  for  small  amplitudes.  It  will,  in  fact, 
be  the  same  as  that  of  a  simple  pendulum  having  the  same  length 
as  the  cycloidal  pendulum  and  vibrating  in  a  small  arc. 

Attempts  have  been  made  to  adapt  the  cycloidal  pendulum  to 
clocks,  but  it  has  been  found  that,  owing  to  the  greater  amount 
of  friction,  its  rate  was  less  regular  than  that  of  the  common  pendu- 
lum. It  may  be  remarked,  that  the  spring  by  which  pendulums  are 
often  suspended  has  the  effect  of  guiding  the  pendulum  bob  in  a 
curve  which  is  approximately  cycloidal,  and  thus  of  diminishing  the 
irregularity  of  rate  resulting  from  differences  of  amplitude. 
,  114.  Moment  of  Inertia. — Just  as  the  mass  of  a  body  is  the 
measure  of  the  force  requisite  for  producing  unit  acceleration  when 
the  movement  is  one  of  pure  translation;  so  the  moment  of  inertia 
of  a  rigid  body  turning  about  a  fixed  axis  is  the  measure  of  the 
couple  requisite  for  producing  unit  acceleration  of  angular  velocity. 

We  suppose  angle  to  be  measured  by  ~j?^  so  that  the  angle  turned 
by  the  body  is  equal  to  the  arc  described  by  any  point  of  it  divided 
by  the  distance  of  this  point  from  the  axis;  and  the  angular  velocity 
of  the  body  will  be  the  velocity  of  any  point  divided  by  its  distance 
from  the  axis.  The  moment  of  inertia  of  the  body  round  the  axis 
is  numerically  equal  to  the  couple  which  would  produce  unit  change 
of  angular  velocity  in  the  body  in  unit  time.  We  shall  now  show 
how  to  express  the  moment  of  inertia  in  terms  of  the  masses  of  the 
particles  of  the  body  and  their  distances  from  the  axis. 

1  Since  the  evolute  of  the  cycloid  is  an  equal  cycloid. 


MOMENT   OF   INERTIA.  69 

Let  m  denote  the  mass  of  any  particle,  r  its  distance  from  the 
axis,  and  <f>  the  angular  acceleration.  Then  r<f>  is  the  acceleration  of 
the  particle  m,  and  the  force  which  would  produce  this  acceleration 
by  acting  directly  on  the  particle  along  the  line  of  its  motion  is 
mr<f>.  The  moment  of  this  force  round  the  axis  would  be  mr?0  since 
its  arm  is  r.  The  aggregate  of  all  such  moments  as  this  for  all  the 
particles  of  the  body  is  evidently  equal  to  the  couple  which  actually 
produces  the  acceleration  of  the  body.  Using  the  sign  2  to  denote 
"  the  sum  of  such  terms  as,"  and  observing  that  $  is  the  same  for  the 
whole  body,  we  have 

Applied  couple  =  2  (mi3<(>)  =  tf>  "2,  (mr2).  (!) 

When  0  is  unity,  the  applied  couple  will  be  equal  to  S  (mrz),  which 
is  therefore,  by  the  foregoing  definition,  the  moment  of  inertia  of 
the  body  round  the  axis. 

-  115.  Moments  of  Inertia  Round  Parallel  Axes.  —  The  moment  of 
inertia  round  an  axis  through  the  centre  of  mass  is  always  less  than 
that  round  any  parallel  axis. 

For  if  r  denote  the  distance  of  the  particle  m  from  an  axis  not 
passing  through  the  centre  of  mass,  and  x  and  y  its  distances  from  two 
mutually  rectangular  planes  through  this  axis,  we  have  rz=x2+y2. 

Now  let  two  planes  parallel  to  these  be  drawn  through  the  centre 
of  mass;  let  I  and  i)  be  the  distances  of  m  from  them,  and  p  its 
distance  from  their  line  of  intersection,  which  will  clearly  be  parallel 
to  the  given  axis.  Also  let  a  and  b  be  the  distances  respectively 
between  the  two  pairs  of  parallel  planes,  so  that  a*+62  will  be  the 
square  of  the  distance  between  the  two  parallel  axes,  which  distance 
we  will  denote  by  h.  Then  we  have 


x2  =  a"  -(-  g2  ±  2a  f  ,  y8  =  6"  +  if  ±  26  17. 

2  (wr2)  =  2  {m  (a8  +  ¥)}  +  2  {m  (f  +  i»')} 

±  2a  2  (m£)  ±  26  2  (mi)) 
=  A2  2?n.  +  I,  (m,p*)  ±  2a  f  2m  ±  26  77  I.m. 

where  ^  and  77  are  the  values  of  £  and  n  for  the  centre  of  mass.  But 
these  values  are  both  zero,  since  the  centre  of  mass  lies  on  both  the 
planes  from  which  £  and  rj  are  measured.  We  have  therefore 

2  (mr*)  =  A2  2m  +  2  (mf),  (2) 

that  is  to  say,  the  moment  of  inertia  round  the  given  axis  exceeds 
the  moment  of  inertia  round  the  parallel  axis  through  the  centre  of 


70  THE   PENDULUM. 

mass  by  the  product  of  the  whole  mass  into  the  square  of  the  dis- 
tance between  the  axes. 

116.  Application  to  Compound  Pendulum.  —  The  application  of  this 
principle  to  the  compound  pendulum  leads  to  some  results  of  great 
interest  and  importance. 

Let  M  be  the  mass  of  a  compound  pendulum,  that  is,  a  rigid  body 
free  to  oscillate  about  a  fixed  horizontal  axis.  Let  h,  as  in  the 
preceding  section,  denote  the  distance  of  the  centre  of  mass  from 
this  axis;  let  0  denote  the  inclination  of  h  to  the  vertical,  and  <f>  the 
angular  acceleration. 

Then,  since  the  forces  of  gravity  on  the  body  are  equivalent  to  a 
single  force  Mg,  acting  vertically  downwards  at  the  centre  of  mass, 
and  therefore  having  an  arm  h  sin  0  with  respect  to  the  axis,  the 
moment  of  the  applied  forces  round  the  axis  is  M#/i  sin  0;  and  this 
must,  by  §  114,  be  equal  to  0S  (rar2).  We  have  therefore 

2  (mr2)       g  sin  6 

-SOT  =  -*- 

If  the  whole  mass  were  collected  at  one  point  at  distance  I  from  the 
axis,  this  equation  would  become 

MZS  g  sin  6, 

Mi  =  l  =  ~^T' 

and  the  angular  motion  would  be  the  same  as  in  the  actual  case  if 
I  had  the  value 

>**-£• 

I   is   evidently  the    length   of   the   equivalent    simple 
pendulum. 

117.  Convertibility  of  Centres.—  Again,  if  we  introduce 
a  length  k  such  that  M&2  is  equal  to  2  (wp2),  that  is,  to 
the  moment  of  inertia  round  a  parallel  axis  through  the 
centre  of  mass,  we  have 


Flg-  48' 


Zro  =  M£z  +  MA», 


and  equation  (5)  becomes 


In  the  annexed  figure  (Fig.  48)  which  represents  a  vertical  section 
through  the  centre  of  mass,  let  G  be  the  centre  of  mass,  A  the  "centre 


COMPOUND   PENDULUM.  71 

of  suspension,"  that  is,  the  point  in  which  the  axis  cuts  the  plane 
of  the  figure,  and  0  the  "  centre  of  oscillation,"  that  is,  the  point  at 
which  the  mass  might  be  collected  without  altering  the  movement. 
Then,  by  definition,  we  have 

I  =  AO,  h  =  AG,  therefore  I- h  =  GO, 

so  that  equation  (7)  signifies 

i*  =  AG  .  GO.  (8) 

Since  k2  is  the  same  for  all  parallel  axes,  this  equation  shows  that 
when  the  body  is  made  to  vibrate  about  a  parallel  axis  through  O, 
the  centre  of  oscillation  will  be  the  point  A.  That  is  to  say;  the 
centres  of  suspension  and  oscillation  are  interchangeable,  and  the 
product  of  their  distances  from  the  centre  of  mass  is  kz. 

118.  If  we  take  a  new  centre  of  suspension  A'  in  the  plane  of  the 
figure,  the  new  centre  of  oscillation  O'  will  lie  in  the  production  of 
A'G,  and  we  must  have 

A'G.GO'  =  P  =  AG.GO. 

If  A'G  be  equal  to  AG,  GO'  will  be  equal  to  GO,  and  A'O'  to  AO, 
so  that  the  length  of  the  equivalent  simple  pendulum  will  be  un- 
changed. A  compound  pendulum  will  therefore  vibrate  in  the 
same  time  about  all  parallel  axes  which  are  equidistant  from  the 
centre  of  mass. 

When  the  product  of  two  quantities  is  given,  their  sum  is  least 
when  they  are  equal,  and  becomes  continually  greater  as  they 
depart  further  from  equality.  Hence  the  length  of  the  equivalent 
simple  pendulum  AO  or  AG  +  GO  is  least  when 

AG  =  GO  =  Jc, 

and  increases  continually  as  the  distance  of  the  centre  of  suspen- 
sion from  G  is  either  increased  from  k  to  infinity  or  diminished  from 
k  to  zero.  Hence,  when  a  body  vibrates  about  an  axis  which  passes 
very  nearly  through  its  centre  of  gravity,  its  oscillations  are  exceed- 
ingly slow. 

s  119.  Kater's  Pendulum. — The  principle  of  the  convertibility  of 
centres,  established  in  §  117,  was  discovered  by  Huygens,  and 
affords  the  most  convenient  practical  method  of  constructing  a 
pendulum  of  known  length.  In  Kater's  pendulum  there  are  two 
parallel  knife-edges  about  either  of  which  the  pendulum  can  be 
made  to  vibrate,  and  one  of  them  can  be  adjusted  to  any  distance 


72  THE   PENDULUM. 

from  the  other.  The  pendulum  is  swung  first  upon  one  of  these 
edges  and  then  upon  the  other,  and,  if  any  difference  is  detected  in 
the  times  of  vibration,  it  is  corrected  by  moving  the  adjustable  edge. 
When  the  difference  has  been  completely  destroyed,  the  distance 
between  the  two  edges  is  the  length  of  the  equivalent  simple  pendu- 
lum. It  is  necessary,  in  any  arrangement  of  this  kind,  that  the  two 
knife-edges  should  be  in  a  plane  passing  through  the  centre  of  gravity; 
also  that  they  should  be  on  opposite  sides  of  the  centre  of  gravity, 
and  at  unequal  distances  from  it. 

120.  Determination  of  the  Value  of  g. — Returning  to  the  formula  for 

the  simple  pendulum  T^v^j,  we  easily  deduce  from  it  gr=|a 
whence  it  follows  that  the  value  of  g  can  be  determined  by  making 
a  pendulum  vibrate  and  measuring  T  and  I.  T  is  determined  by 
counting  the  number  of  vibrations  that  take  place  in  a  given  time; 
I  can  be  calculated,  when  the  pendulum  is  of  regular  form,  by  the 
aid  of  formulse  which  are  given  in  treatises  on  rigid  dynamics,  but 
its  value  is  more  easily  obtained  by  Rater's  method,  described  above, 
founded  on  the  principle  of  the  convertibility  of  the  centres  of 
suspension  and  oscillation. 

It  is  from  pendulum  observations,  taken  in  great  numbers  at 
different  parts  of  the  earth,  that  the  approximate  formula  for  the 
intensity  of  gravity  which  we  have  given  at  §  91  has  been  deduced. 
Local  peculiarities  prevent  the  possibility  of  laying  down  any  general 
formula  with  precision;  and  the  exact  value  of  g  for  any  place  can 
only  be  ascertained  by  observations  on  the  spot. 


CHAPTER  IX. 


CONSERVATION   OF   ENERGY. 


121.  Definition  of  Kinetic  Energy.— We  have  seen  in  §  93  that  the 
work  which  must  be  done  upon  a  mass  of  m  grammes  to  give  it  a 
velocity  of  v  centimetres  per  second  is  |rai>2  ergs.      Though  we  have 
proved  this  only  for  the  case  of  falling  bodies,  with  gravity  as  the 
working  force,  the  result  is  true  universally,  as  is  shown  in  advanced 
treatises  on  mathematical  physics.     It  is  true  whether  the  motion 
be  rectilinear  or  curvilinear,  and  whether  the  working  force  act  in 
the  line  of  motion  or  at  an  angle  with  it. 

If  the  velocity  of  a  mass  increases  from  v-^  to  v2,  the  work  done 
upon  it  in  the  interval  is  |m  (v2z  —  va2);  in  other  words,  is  the 
increase  of  |mv2. 

On  the  other  hand,  if  a  force  acts  in  such  a  manner  as  to  oppose 
the  motion  of  a  moving  mass,  the  force  will  do  negative  work,  the 
amount  of  which  will  be  equal  to  the  decrease  in  the  value  of  \mvz. 

For  example,  during  any  portion  of  the  ascent  of  a  projectile,  the 
diminution  in  the  value  of  %mv2  is  equal  to  gm  multiplied  by  the 
increase  of  height;  and  during  any  portion  of  its  descent  the  increase 
in  ^mvz  is  equal  to  gm  multiplied  by  the  decrease  of  height. 

The  work  which  must  have  been  done  upon  a  body  to  give  it  its 
actual  motion,  supposing  it  to  have  been  initially  at  rest,  is  called 
the  energy  of  motion  or  the  kinetic  energy  of  the  body.  It  can  be 
computed  by  multiplying  half  the  mass  by  the  square  of  the  velocity. 

122.  Definition  of  Static  or  Potential  Energy. — When  a  body  of 
mass  m  is  at  a  height  s  above  the  ground,  which  we  will  suppose 
level,  gravity  is  ready  to  do  the  amount  of  work  gms  upon  it  by 
making  it  fall  to  the  ground.     A  body  in  an  elevated  position  may 
therefore  be  regarded  as  a  reservoir  of  work.     In  like  manner  a 
wound-up  clock,  whether  driven  by  weights  or  by  a  spring,  has 


74  CONSERVATION    OF   ENERGY. 

work  stored  up  in  it.  In  all  these  cases  there  is  force  between  parts 
of  a  system  tending  to  produce  relative  motion,  and  there  is  room 
for  such  relative  motion  to  take  place.  There  is  force  ready  to  act, 
and  space  for  it  to  act  through.  Also  the  force  is  always  the  same 
in  the  same  relative  position  of  the  parts.  Such  a  system  possesses 
energy,  which  is  usually  called  potential  We  prefer  to  call  it 
statical,  inasmuch  as  its  amount  is  computed  on  statical  principles 
alone.1  Statical  energy  depends  jointly  on  mutual  force  and  relative 
position.  Its  amount  in  any  given  position  is  the  amount  of  work 
which  would  be  done  by  the  forces  of  the  system  in  passing  from 
this  position  to  the  standard  position.  When  we  are  speaking  of 
the  energy  of  a  heavy  body  in  an  elevated  position  above  level 
ground,  we  naturally  adopt  as  the  standard  position  that  in  which 
the  body  is  lying  on  the  ground.  When  we  speak  of  the  energy  of 
a  wound-up  clock,  we  adopt  as  the  standard  position  that  in  which 
the  clock  has  completely  run  down.  Even  when  the  standard 
position  is  not  indicated,  we  can  still  speak  definitely  of  the  differ- 
ence between  the  energies  of  two  given  positions  of  a  system;  just 
as  we  can  speak  definitely  of  the  difference  of  level  of  two  given 
points  without  any  agreement  as  to  the  datum  from  which  levels 
are  to  be  reckoned. 

123.  Conservation  of  Mechanical  Energy. — When  a  frictionless 
system  is  so  constituted  that  its  forces  are  always  the  same  in  the 
same  positions  of  the  system,  the  amount  of  work  done  by  these 
forces  during  the  passage  from  one  position  A  to  another  position  B 
will  be  independent  of  the  path  pursued,  and  will  be  equal  to  minus 
the  work  done  by  them  in  the  passage  from  B  to  A.  The  earth  and 
any  heavy  body  at  its  surface  constitute  such  a  system;  the  force  of  the 
system  is  the  mutual  gravitation  of  these  two  bodies;  and  the  work 
done  by  this  mutual  gravitation,  when  the  body  is  moved  by  any 
path  from  a  point  A  to  a  point  B,  is  equal  to  the  weight  of  the  body 
multiplied  by  the  height  of  A  above  B.  When  the  system  passes 
through  any  series  of  movements  beginning  with  a  given  position 
and  ending  with  the  same  position  again,  the  algebraic  total  of  work 
done  by  the  forces  of  the  system  in  this  series  of  movements  is  zero. 
For  instance,  if  a  heavy  body  be  carried  by  a  roundabout  path  back 
to  the  point  from  whence  it  started,  no  work  is  done  upon  it  by 
gravity  upon  the  whole. 

Every  position  of  such  a  system  has  therefore  a  definite  amount 

1  That  is  to  say,  the  computation  involves  no  reference  to  the  laws  of  motion. 


TRANSFORMATIONS    OF   ENERGY.  75 

of  statical  energy,  reckoned  with  respect  to  an  arbitrary  standard 
position.  The  work  done  by  the  forces  of  the  system  in  passing 
from  one  position  to  another  is  (by  definition)  equal  to  the  loss  of 
static  energy;  but  this  loss  is  made  up  by  an  equal  gain  of  kinetic 
energy.  Conversely  if  kinetic  energy  is  lost  in  passing  from  one 
position  to  another,  the  forces  do  negative  work  equal  to  this  loss, 
and  an  equal  amount  of  static  energy  is  gained.  The  total  energy 
of  the  system  (including  both  static  and  kinetic)  therefore  remains 
unaltered. 

An  approximation  to  such  a  state  of  things  is  exhibited  by  a 
pendulum.  In  the  two  extreme  positions  it  is  at  rest,  and  has  there- 
fore no  kinetic  energy;  but  its  statical  energy  is  then  a  maximum. 
In  the  lowest  position  its  motion  is  most  rapid;  its  kinetic  energy  is 
therefore  a  maximum,  but  its  statical  energy  is  zero.  The  difference 
of  the  statical  energies  of  any  two  positions,  will  be  the  weight  of 
the  pendulum  multiplied  by  the  difference  of  levels  of  its  centre  of 
gravity,  and  this  will  also  be  the  difference  (in  inverse  order)  between 
the  kinetic  energies  of  the  pendulum  in  these  two  positions. 

As  the  pendulum  is  continually  setting  the  air  in  motion  and  thus 
doing  external  work,  it  gradually  loses  energy  and  at  last  comes  to 
rest,  unless  it  be  supplied  with  energy  from  a  clock  or  some  other 
source.  If  a  pendulum  could  be  swung  in  a  perfect  vacuum,  with 
an  entire  absence  of  friction,  it  would  lose  no  energy,  and  would 
vibrate  for  an  indefinite  time  without  decrease  of  amplitude. 

124.  Illustration  from  Pile-driving. — An  excellent  illustration  of 
transformations  of   energy  is   furnished   by  pile-driving.     A  large 
mass  of  iron  called  a  ram  is  slowly  hauled  up  to  a  height  of  several 
yards  above  the  pile,  and  is  then  allowed  to  fall  upon  it.     During 
the  ascent,  work  must  be  supplied  to  overcome  the  force  of  gravity; 
and  this  work  is  represented  by  the  statical  energy  of  the  ram  in  its 
highest  position.     While  falling,  it  continually  loses  statical  and 
gains  kinetic  energy;  the  amount  of  the  latter  which  it  possesses 
immediately  before  the  blow  being  equal  to  the  work  which  has 
been  done  in  raising  it.     The  effect  of  the  blow  is  to  drive  the  pile 
through  a  small  distance  against  a  resistance  very  much  greater  than 
the  weight  of  the  ram;  the  work  thus  done  being  nearly  equal  to 
the  total  energy  which  the  ram  possessed  at  any  point  of  its  descent. 
We  say  nearly  equal,  because  a  portion  of  the  energy  of  the  blow  is 
spent  in  producing  vibrations. 

125.  Hindrances    to   Availability    of    Energy. — There    is    almost 


70  CONSERVATION   OF   ENERGY. 

always  some  waste  in  utilizing  energy.  When  water  turns  a  mill- 
wheel,  it  runs  away  from  the  wheel  with  a  velocity,  the  square  of 
which  multiplied  by  half  the  mass  of  the  water  represents  energy 
which  has  run  to  waste. 

Friction  again  often  consumes  a  large  amount  of  energy;  and  in 
this  case  we  cannot  (as  in  the  preceding  one)  point  to  any  palpable 
motion  of  a  mass  as  representing  the  loss.  Heat,  however,  is  pro- 
duced, and  the  energy  which  has  disappeared  as  regarded  from  a 
gross  mechanical  point  of  view,  has  taken  a  molecular  form.  Heat 
is  a  form  of  molecular  energy;  and  we  know,  from  modern  re- 
searches, what  quantity  of  heat  is  equivalent  to  a  given  amount  of 
mechanical  work.  In  the  steam-engine  we  have  the  converse 
process;  mechanical  work  is  done  by  means  of  heat,  and  heat  is 
destroyed  in  the  doing  of  it,  so  that  the  amount  of  heat  given  out 
by  the  engine  is  less  than  the  amount  supplied  to  it. 

The  sciences  of  electricity  and  magnetism  reveal  the  existence  of 
other  forms  of  molecular  energy;  and  it  is  possible  in  many  ways  to 
produce  one  form  of  energy  at  the  expense  of  another;  but  in  every 
case  there  is  an  exact  equivalence  between  the  quantity  of  one  kind 
which  comes  into  existence  and  the  quantity  of  another  kind  which 
simultaneously  disappears.  Hence  the  problem  of  constructing  a 
self-driven  engine,  which  we  have  seen  to  be  impossible  in  mechanics, 
is  equally  impossible  when  molecular  forms  of  energy  are  called  to 
the  inventor's  aid. 

Energy  may  be  transformed,  and  may  be  communicated  from  one 
system  to  another;  but  it  cannot  be  increased  or  diminished  in  total 
amount.  This  great  natural  law  is  called  the  principle  of  the  con- 
servation of  energy. 


CHAPTER   X. 


ELASTICITY. 


126.  Elasticity  and  its  Limits. — There  is  no  such  thing  in  nature 
as  an  absolutely  rigid  body.  All  bodies  yield  more  or  less  to  the 
action  of  force;  and  the  property  in  virtue  of  which  they  tend  to 
recover  their  original  form  and  dimensions  when  these  are  forcibly 
changed,  is  called  elasticity.  Most  solid  bodies  possess  almost  per- 
fect elasticity  for  small  deformations;  that  is  to  say,  when  distorted, 
extended,  or  compressed,  within  certain  small  limits,  they  will,  on 
the  removal  of  the  constraint  to  which  they  have  been  subjected, 
instantly  regain  almost  completely  their  original  form  and  dimen- 
sions. These  limits  (which  are  called  the  limits  of  elasticity)  are 
different  for  different  substances;  and  when  a  body  is  distoited 
beyond  these  limits,  it  takes  a  set,  the  form  to  which  it  returns 
being  intermediate  between  its  original  form  and  that  into  which  it 
was  distorted. 

When  a  body  is  distorted  within  the  limits  of  its  elasticity,  the 
force  with  which  it  reacts  is  directly  proportional  to  the  amount  of 
distortion.  For  example,  the  force  required  to  make  the  prongs  of 
a  tuning-fork  approach  each  other  by  a  tenth  of  an  inch,  is  double 
of  that  required  to  produce  an  approach  of  a  twentieth  of  an  inch; 
and  if  a  chain  is  lengthened  a  twentieth  of  an  inch  by  a  weight  of 
1  cwt.,  it  will  be  lengthened  a  tenth  of  an  inch  by  a  weight  of  2 
cwt.,  the  chain  being  supposed  to  be  strong  enough  to  experience  no 
permanent  set  from  this  greater  weight.  Also,  within  the  limits  of 
elasticity,  equal  and  opposite  distortions,  if  small,  are  resisted  by 
equal  reactions.  For  example,  the  same  force  which  suffices  to 
make  the  prongs  of  a  tuning-fork  approach  by  a  twentieth  of  an 
inch,  will,  if  applied  in  the  opposite  direction,  make  them  separate 
by  the  same  amount. 


78 


ELASTICITY. 


127.  Isochronism  of  Small  Vibrations.— An  important  consequence 
of   these  laws  is,  that  when  a  body  receives  a  slight  distortion 
within  the  limits  of  its  elasticity,  the  vibrations  which  ensue  when 
the  constraint  is  removed  are  isochronous.     This  follows  from  §  111, 
in'asmuch  as  the  accelerations  are  proportional  to  the  forces,  and  are 
therefore  proportional  at  each  instant  to  the  deformation  at  that 
instant. 

128.  Stress,  Strain,  and  Coefficients  of  Elasticity.— A  body  which, 
like  indian-rubber,  can  be  subjected  to  large  deformations  without 
receiving  a  permanent  set,  is  said  to  have  wide  limits  of  elasticity. 

A  body  which,  like  steel,  opposes  great  resistance  to  deformation, 
is  said  to  have  large  coefficients  of  elasticity. 

Any  change  in  the  shape  or  size  of  a  body  produced  by  the  appli- 
cation of  force  to  the  body  is  called  a  strain;  and  an  action  of  force 
tending  to  produce  a  strain  is  called  a  stress. 

When  a  wire  of  cross-section  A  is  stretched  with  a  force  F,  the 
longitudinal  stress  is  -r;  this  being  the  intensity  of  force  per  unit 
area  with  which  the  two  portions  of  the  wire  separated  by  any 
cross-section  are  pulling  each  other.  If  the  length  of  the  wire  when 
unstressed  is  L  and  when  stressed  L+Z,  the  longitudinal  strain  is 

L.  A  stress  is  always  expressed  in  units  of  force  per  unit  of  area. 
A  strain  is  always  expressed  as  the  ratio  of  two  magnitudes  of  the 
same  kind  (in  the  above  example,  two  lengths),  and  is  therefore 
independent  of  the  units  employed. 

The  quotient  of  a  stress  by  the  strain  (of  a  given  kind)  which  it 
produces,  is  called  a  coefficient  or  modulus  of  elasticity.  In  the  above 
example,  the  quotient  —^  is  called  Young's  modulus  of  elasticity. 

As  the  wire,  while  it  extends  lengthwise,  contracts  laterally,  there 
will  be  another  coefficient  of  elasticity  obtained  by  dividing  the 
longitudinal  stress  by  the  lateral  strain. 

It  is  shown,  in  special  treatises,  that  a  solid  substance  may  have 
21  independent  coefficients  of  elasticity;  but  that  when  the  substance 
is  isotropic,  that  is,  has  the  same  properties  in  all  directions,  the 
number  reduces  to  2. 

129.  Volume-elasticity.— The  only  coefficient  of  elasticity  possessed 
by  liquids  and  gases  is  elasticity  of  volume.     When  a  body  of  volume 
V  is  reduced  by  the  application  of  uniform  normal  pressure  over  its 
whole  surface  to  volume  V—v,  the  volume-strain  is  -,  and  if  this 


COEFFICIENTS   OF   ELASTICITY. 


79 


effect  is  produced  by  a  pressure  of  p  units  of  force  per  unit  of  area, 
the  elasticity  of  volume  is  the  quotient  of  the  stress  p  by  the  strain 

^,  or  is  — .     This  is  also  called  the  resistance  to  compression; 

and  its  reciprocal  -y  is  called  the  compressibility  of  the  substance. 

In  dealing  with  gases,  p  must  be  understood  as  a  pressure  super- 
added  to  the  original  pressure  of  the  gas. 

Since  a  strain  is  a  mere  numerical  quantity,  independent  of  units, 
a  coefficient  of  elasticity  must  be  expressed,  like  a  stress,  in  units  of 
force  per  unit  of  area.  In  the  C.G.S.  system,  stresses  and  coefficients 
of  elasticity  are  expressed  in  dynes  per  square  centimetre.  The 
following  are  approximate  values  (thus  expressed)  of  the  two  co- 
efficients of  elasticity  above  denned: — 


Voung's 

Elasticity  of 

Modulus. 

Volume. 

Glass  (flint), 

60  x  1010 

40  x  1010 

Steel, 

210  x  10»° 

ISO  x  10l° 

Iron  (wrought), 

190  x  1010 

140  x  10M 

Iron  (cast), 

130  x  10l° 

96  x  10l° 

Copper, 

120  x  1010 

160  x  10W 

Mercury, 

54  x  1010 

Water, 

2  x  1010 

Alcohol, 

1-2  x  1010 

130.  (Ersted's  Piezometer.— The 
compression  of  liquids  has  been 
observed  by  means  of  (Ersted's 
piezometer,  which  is  represented 
in  Fig.  49.  The  liquid  whose 
compression  is  to  be  observed  is 
contained  in  a  glass  vessel  b, 
resembling  a  thermometer  with 
a  very  large  bulb  and  short  tube. 
The  tube  is  open  above*  and  a 
globule  of  mercury  at  the  top 
of  the  liquid  column  serves  as  an 
index.  This  apparatus  is  placed 
in  a  very  strong  glass  vessel  a  full 
of  water.  When  pressure  is  exerted  by  means  of  the  piston  Wi, 
the  index  of  mercury  is  seen  to  descend,  showing  a  diminution  of 
volume  of  the  liquid,  and  showing  moreover  that  this  diminution  of 
volume  exceeds  that  of  the  containing  vessel  b.  It  might  at  first 


Fig.  49.-<Et8teds  Piezometer: 


80  ELASTICITY. 

sight  appear  that  since  this  vessel  is  subjected  to  equal  pressure 
within  and  without,  its  volume  is  unchanged;  but  in  fact,  its 
volume  is  altered  to  the  same  extent  as  that  of  a  solid  vessel  of  the 
same  material;  for  the  interior  shells  would  react  with  a  force 
precisely  equivalent  to  that  which  is  exerted  by  the  contained 
liquid. 


CHAPTER  XI. 


FRICTION. 


131.  Friction,  Kinetical  and  Statical. — When  two  bodies  are  pressed 
together  in  such  a  manner  that  the  direction  of  their  mutual  pressure 
is  not  normal  to  the  surface  of  contact,  the  pressure  can  be  resolved 
into  two  parts,  one  normal  and  the  other  tangential.     The  tangential 
component  is  called  the  force  of  friction  between  the  two  bodies. 
The  friction  is  called  kinetical  or  statical  according  as  the  bodies 
are  or  are  not  sliding  one  upon  the  other. 

As  regards  kinetical  friction,  experiment  shows  that  if  the  normal 
pressure  between  two  given  surfaces  be  changed,  the  tangential  force 
changes  almost  exactly  in  the  same  proportion;  in  other  words,  the 
ratio  of  the  force  of  friction  to  the  normal  pressure  is  nearly  constant 
for  two  given  surfaces.  This  ratio  is  called  the  coefficient  of  kinetical 
friction  between  the  two  surfaces,  and  is  nearly  independent  of  the 
velocity. 

132.  Statical  Friction.     Limiting  Angle. — It  is  obvious  that  the 
statical  friction  between  two  given  surfaces  is  zero  when  their  mutual 
pressure  is  normal,  and  increases  with  the  obliquity  of  the  pressure 
if  the  normal  component  be  preserved  constant.      The  obliquity, 
however,  cannot  increase  beyond  a  certain  limit,  depending  on  the 
nature  of  the  bodies,  and  seldom  amounting  to  so  much  as  45°.     Be- 
yond this  limit  sliding  takes  place.     The  limiting  obliquity,  that  is, 
the  greatest  angle  that  the  mutual  force  can  make  with  the  normal, 
is  called  the  limiting  angle  of  friction  for  the  two  surfaces;  and 
the  ratio  of  the  tangential   to  the  normal  component  when  the 
mutual  force  acts  at  the  limiting  angle,  is  called  the  coefficient  of 
statical  friction  for  the  two  surfaces.     The  coefficient  and  limiting 
angle  remain  nearly  constant  when  the  normal  force  is  varied. 

The  coefficient  of  statical  friction  is  in  almost  every  case  greater 


82  FRICTION. 

than  the  coefficient  of  kinetical  friction;  in  other  words,  friction 
offers  more  resistance  to  the  commencement  of  sliding  than  to  the 
continuance  of  it. 

A  body  which  has  small  coefficients  of  friction  with  other  bodies 
is  called  slippery. 

133.  Coefficient^ tan  6.  Inclined  Plane. — If  0  be  the  inclination 
of  the  mutual  force  P  to  the  common  normal,  the  tangential  com- 
ponent will  be  P  sin  6,  the  normal  component  P  cos  6,  and  the  ratio 
of  the  former  to  the  latter  will  be  tan  0.  Hence  the  coefficient  of 
statical  friction  is  equal  to  the  tangent  of  the  limiting  angle  of 
friction.  , 

When  a  heavy  body  rests  on  an  inclined  plane,  the  mutual  pressure 
is  vertical,  and  the  angle  8  is  the  same  as  the  inclination  of  the 
plane.  Hence  if  an  inclined  plane  is  gradually  tilted  till  a  body 
lying  on  it  slides  under  the  action  of  gravity,  the  inclination  of  the 
plane  at  which  sliding  begins  is  the  limiting  angle  of  friction 
between  the  body  and  the  plane,  and  the  tangent  of  this  angle  is  the 
coefficient  of  statical  friction. 

Again,  if  the  inclination  of  a  plane  be  such  that  the  motion  of  a 
body  sliding  down  it  under  the  action  of  gravity  is  neither  accelerated 
nor  retarded,  the  tangent  of  this  inclination  will  be  the  coefficient 
of  kinetical  friction. 


CHAPTER   XII. 


HYDROSTATICS. 


-  134.  Hydrodynamics. — We  shall  now  treat  of  the  laws  of  force  as 
applied  to  fluids.  This  branch  of  the  general  science  of  dynamics  is 
called  hydrodynamics  (vdup,  water),  and  is  divided  into  hydrostatics 
and  hydrokinetics.  Our  discussions  will  be  almost  entirely  confined 
to  hydrostatics.  , 

FLUIDS. — TRANSMISSION   OF  PRESSURE. 
The  name  fluid  comprehends  both  liquids  and  gases. 

- 135.  No  Statical  Friction  in  Fluids. — A  fluid  at  rest  cannot  exert 
any  tangential  force  against  a  surface  in  contact  with  it;  its  pressure 
at  every  point  of  such  a  surface  is  entirely  normal.  A  slight  tangen- 
tial force  is  exerted  by  fluids  in  motion;  and  this  fact  is  expressed 
by  saying  that  all  fluids  are  more  or  less  viscous.  An  imaginary 
perfect  fluid  would  be  perfectly  free  from  viscosity;  its  pressure 
against  any  surface  would  be  entirely  normal,  whether  the  fluid 
were  in  motion  or  at  rest. 

,136.  Intensity  of  Pressure. — When  pressure  is  uniform  over  an 
area,  the  total  amount  of  the  pressure,  divided  by  the  area,  is  called 
the  intensity  of  the  pressure.  The  C.G.S.  unit  of  intensity  of 
pressure  is  a  pressure  of  a  dyne  on  each  square  centimetre  of  sur- 
face. A  rough  unit  of  intensity  frequently  used  is  the  pressure  of 
a  pound  per  square  inch.  This  unit  varies  with  the  intensity  of 
gravity,  and  has  an  average  value  of  about  69,000  C.G.S.  unite. 
Another  rough  unit  of  intensity  of  pressure  frequently  employed  is 
"  an  atmosphere  " — that  is  to  say,  the  average  intensity  of  pressure 
of  the  atmosphere  at  the  surface  of  the  earth.  This  is  about 
1,000,000  C.G.S.  units.  .  ' 


84  HYDROSTATICS. 

The  single  word  "  pressure  "  is  used  sometimes  to  denote  "  amount 
of  pressure"  (which  can  be  expressed  in  dynes)  and  sometimes 
"  intensity  of  pressure"  (which  can  be  expressed  in  dynes  per  square 
centimetre).  The  context  usually  serves  to  show  which  of  these 
two  meanings  is  intended. 

137.  Pressure  the  Same  in  all  Directions. — The  intensity  of  pressure 
at  any  point  of  a  fluid  is  the  same  in  all  directions;  it  is  the  same 
whether  the  surface  which  receives  the  pressure  faces  upwards, 
downwards,  horizontally,  or  obliquely. 

This  equality  is  a  direct  consequence  of  the  absence  of  tangential 
force  between  two  contiguous  portions  of  a  fluid. 

For  in  order  that  a  small  triangular  prism  of  the  fluid  (its  ends 
being  right  sections)  may  be  in  equilibrium,  the  pressures  on  its 
three  faces  must  balance  each  other.  But  when  three  forces  balance 
each  other,  they  are  proportional  to  the  sides  of  a  triangle  to  which 
they  are  perpendicular;1  hence  the  amounts  of  pressure  on  the 
three  faces  are  proportional  to  the  faces,  in  other  words  the  inten- 
sities of  these  three  pressures  are  equal.  As  we  can  take  two  of 
the  faces  perpendicular  to  any  two  given  directions,  this  proves  that 
the  pressures  in  all  directions  at  a  point  are  of  equal  intensity. 

138.  Pressure  the  Same  at  the  Same  Level. — 
In  a  fluid  at  rest,  the  pressure  is  the  same 
at  all  points  in  the  same  horizontal  plane. 
This  appears  from  considering  the  equilibrium 
of  a  horizontal  cylinder  AB  (Fig.  50),  of  small 
sectional  area,  its  ends  being  right  sections. 
The  pressures  on  the  sides  are  normal,  and 
therefore  give  no  component  in  the  direction 
of  the  length;  hence  the  pressures  on  the 

ends  must  be  equal  in  amount ;  but  they  act  on  equal  areas;  there- 
fore their  intensities  are  equal. 

A  horizontal  surface  in  a  liquid  at  rest  may  therefore  be  called  a 
"  surface  of  equal  pressure." 

-  139.  Difference  of  Pressure  at  Different  Levels.— The  increase  of 
pressure  with  depth,  in  a  fluid  of  uniform  density,  can  be  investi- 
gated as  follows:— Consider  the  equilibrium  of  a  vertical  cylinder 
mm'  (Fig.  51),  its  ends  being  right  sections.  The  pressures  on  its 

1  This  is  an  obvious  consequence  of  the  triangle  of  forces  (art.  14);  for  if  the  sides  of 
a  triangle  are  parallel  to  three  forces,  we  have  only  to  turn  the  triangle  through  a  right 
angle,  and  its  sides  will  then  be  perpendicular  to  the  forces. 


INCREASE    OF   PRESSURE   WITH    DEPTH. 


85 


sides  are  normal,  and  therefore  horizontal.     The  only  vertical  forces 

acting  upon  it  are  its  own  weight  and  the  pressures  on  its  ends,  of 

which  it  is  to  be  observed  that  the  pressure 

on  the  upper  end  acts  downwards  and  that 

on  the  lower  end  upwards.     The  pressure  on 

the  lower  end  therefore  exceeds  that  on  the 

upper  end  by  an  amount  equal  to  the  weight 

of  the  cylinder.     If  a  be  the  sectional  area,  w 

the  weight  of  unit  volume  of  the  liquid,  and 

h  the  length  of  the  cylinder,  the  volume  of 

the  cylinder  is  ha,  and  its  weight  wha,  which  Fig- 51- 

must  be  equal  to  (p—p)  a  if  p,p'  are  the  intensities  of  pressure  on 

the  lower  and  upper  ends  respectively.     We  have  therefore 


that  is,  the  increase  of  pressure  in  descending  through  a  depth  h 
is  wh. 

The  principles  of  this  and  the  preceding  section  remain  appli- 
cable whatever  be  the  shape  of  the  containing  vessel,  even  if  it  be 
such  as  to  render  a  circuitous  route  necessary  in  passing  from  one 
of  two  points  compared  to  the  other;  for  this  route  can  always  be 
made  to  consist  of  a  succession  of  vertical  and  horizontal  lines,  and 
the  preceding  principles  when  applied  to  each  of  these  lines  separ- 
ately, will  give  as  the  final  result  a  difference  of  pressure  wh  for  a 
difference  of  heights  h. 

If  d  denote  the  density  of  the  liquid,  in  grammes  per  sq.  cm.,  the 
weight  of  a  cubic  cm.  will  be  gd  dynes.  The  increase  of  pressure 
for  an  increase  of  depth  h  cm.  is  therefore  ghd  dynes  per  sq.  cm. 
If  there  be  no  pressure  at  the  surface  of  the  liquid,  this  will  be  the 
actual  pressure  at  the  depth  h. 

^  140.  Free  Surface. — It  follows  from  these  principles  that  the  free 
surface  of  a  liquid  at  rest — that  is,  the 
surface  in  contact  with  the  atmosphere 
— must  be  horizontal;  since  all  points  in 
this  surface  are  at  the  same  pressure.  If 
the  surface  were  not  horizontal,  but  were 
higher  at  n  than  at  n  (Fig.  52),  the  pres- 
sures at  the  two  points  m,  m'  vertically 
beneath  them  in  any  horizontal  plane 
AB  would  be  unequal,  for  they  would  be  due  to  the  weights 


Fig.  52. 


80 


HYDROSTATICS. 


Fig  53. 


of  unequal  columns  nm,  rim,  and  motion  would  ensue  from  m 
towards  m'. 

The  same  conclusion  can  be  deduced  from  considering  the  equili- 
brium of  a  particle  at  the  surface,  as  M  (Fig.  53).     If  the  tangent 
plane  at  M  were  not  horizontal  there  would  be  a  component  of 
gravity  tending  to  make  the  particle 
slide  down;  and  this  tendency  would 
produce  motion,  since  there  is  no  fric- 
tion to  oppose  it. 

141.  Transmissibility  of  Pressure  in 
Fluids. — Since  the  difference  of  the 
pressures  at  two  points  in  a  fluid  can 
be  determined  by  the  foregoing  prin- 
ciples, independently  of  any  knowledge  of  the  absolute  intensity 
of  either,  it  follows  that  when  increase  or  diminution  of  pres- 
sure occurs  at  one  point,  an  equal  increase  or  diminution  must 
occur  throughout  the  whole  fluid.  A  fluid  in  a  closed  vessel 
perfectly  transmits  through  its  whole  substance  whatever  pressure 
we  apply  to  any  part.  The  changes  in  amount  of  pressure  will  be 
equal  for  all  equal  areas.  For  unequal  areas  they  will  be  propor- 
tional to  the  areas. 

Thus  if  the  two  vertical  tubes  in  Fig.  54  have  sectional  areas 
which  are  as  1  to  16,  a  weight  of  1  kilo- 
gram acting  on  the  surface  of  the  liquid 
in  the  smaller  tube  will  be  balanced  by 
16  kilograms  acting  on  the  surface  of  the 
liquid  in  the  larger. 

This  principle  of  the  perfect  transmis- 
sion of  pressure  by  fluids  appears  to  have 
been  first  discovered  and  published  by 
Stevinus;  but  it  was  rediscovered  by 
Pascal  a  few  years  later,  and  having  been 
made  generally  known  by  his  writings  is 
rig.  M.-iwndpieof  the  Hydraulic  of  ten  called  "  Pascal's  principle."  In  his 
celebrated  treatise  on  the  Equilibrium  of 

Liquids,  he  says,  "If  a  vessel  full  of  water,  closed  on  all  sides,  has 
two  openings,  the  one  a  hundred  times  as  large  as  the  other,  and  if 
each  be  supplied  with  a  piston  which  fits  exactly,  a  man  pushing 
the  small  piston  will  exert  a  force  which  will  equilibrate  that  of  a 
hundred  men  pushing  the  piston  which  is  a  hundred  times  as  large, 


16 1L 


TRANSMISSIBILITY   OF   PRESSURE.  87 

and  will  overcome  that  of  ninety-nine.  And  whatever  may  be  the 
proportion  of  these  openings,  if  the  forces  applied  to  the  pistons  are 
to  each  other  as  the  openings,  they  will  be  in  equilibrium." 

142.  Hydraulic  Press. — This  mode  of  multiplying  force  remained 
for  a  long  time  practically  unavailable  on  account  of  the  difficulty 
of  making  the  pistons  water-tight.  The  hydraulic  press  was  first 
successfully  made  by  Bramah,  who  invented  the  cupped  leather  collar 
illustrated  in  Fig.  166,  §  264.  Fig.  165  shows  the  arrangements  of 
the  press  as  a  whole.  Instead  of  pistons,  plungers  are  employed; 
that  is  to  say,  solid  cylinders  of  metal  which  can  be  pushed  down 
into  the  liquid,  or  can  be  pushed  up  by  the  pressure  of  the  liquid 
against  their  bases.  The  volume  of  liquid  displaced  by  the  advance 
of  a  plunger  is  evidently  equal  to  that  displaced  by  a  piston  of  the 
same  sectional  area,  8,nd  the  above  calculations  for  pistons  apply  to 
plungers  as  well.  The  plungers  work  through  openings  which  are 
kept  practically  water-tight  by  means  of  the  cup-leather  arrange- 
ment. The  cup-leather,  which  is  shown  both  in  plan  and  section 
in  Fig.  166,  consists  of  a  leather  ring  bent  so  as  to  have  a  semi- 
circular section.  It  is  fitted  into  a  hollow  in  the  interior  of  the 
sides  of  the  opening,  so  that  water  leaking  up  along  the  circumfer- 
ence of  the  plunger  will  fill  the  concavity  of  the  leather,  and,  by 
pressing  on  it,  will  produce  a  packing  which  fits  more  tightly  as  the 
pressure  on  the  plunger  increases. 

.  143.  Principle  of  Work  Applicable. — In  Fig.  54,  when  the  smaller 
piston  advances  and  forces  the  other  back,  the  volume  of  liquid 
driven  out  of  the  smaller  tube  is  equal  to  the  sectional  area  multi- 
plied by  the  distance  through  which  the  piston  advances.  In  like 
manner,  the  volume  of  liquid  driven  into  the  larger  tube  is  equal  to 
its  sectional  area  multiplied  by  the  distance  that  its  piston  is  forced 
back.  But  these  two  volumes  are  equal,  since  the  same  volume  of 
liquid  that  leaves  one  tube  enters  the  other.  The  distances  through 
which  the  two  pistons  move  are  therefore  inversely  as  their  sectional 
areas,  and  hence  are  inversely  as  the  amounts  of  pressure  applied 
to  them.  The  ^uork  done  in  pushing  forward  the  smaller  piston  is 
therefore  equal  to  the  work  done  by  the  liquid  in  pushing  back  the 
larger.  This  was  remarked  by  Pascal,  who  says — 

"It  is,  besides,  worthy  of  admiration  that  in  this  new  machine 
we  find  that  constant  rule  which  is  met  with  in  all  the  old  ones 
such  as  the  lever,  wheel  and  axle,  screw,  &c.,  which  is  that  the 
distance  is  increased  in  proportion  to  the  force;  for  it  is  evident  that 


88 


HYDROSTATICS 


as  one  of  these  openings  is  a  hundred  times  as  large  as  the  other,  if 
the  man  who  pushes  the  small  piston  drives  it  forward  one  inch,  he 
will  drive  the  large  piston  backward  only  one-hundredth  part  of 
that  length." 

144.  Experiment  on   Upward    Pressure.— The   upward    pressure 

exerted  by  a  liquid  against  a 
horizontal  surface  facing  down- 
wards can  be  exhibited  by  the 
following  experiment.  Take  a  tube 
open  at  both  ends  (Fig.  55),  and 
keeping  the  lower  end  covered 
with  a  piece  of  card,  plunge  it  into 
water.  The  liquid  will  press  the 
card  against  the  bottom  of  the 
tube  with  a  force  which  increases 
as  it  is  plunged  deeper.  If  water 
be  now  poured  into  the  tube,  the 
card  will  remain  in  its  place .  as 
long  as  the  level  of  the  liquid  is 
lower  within  the  tube  than  with- 
out; but  at  the  moment  when 
equality  of  levels  is  attained  it 
will  become  detached. 

145.  Liquids  in  Superposition. — When  one  liquid  rests  on  the  top 
of  another  of  different  density,  the  foregoing  principles  lead  to  the 
result  that  the  surface  of  demarcation  must  be  horizontal.     For  the 
free  surface  of  the  upper  liquid  must,  as  we  have  seen,  be  horizontal. 
If  now  we  take  two  small  equal  areas  n  and  n'  (Fig.  5G)  in  a 
horizontal  layer  of  the  lower   liquid,  they  must  be  subjected  to 

equal  pressures.  But  these  pressures  are 
measured  by  the  weights  of  the  liquid 
cylinders  nrs,  nil;  and  these  latter  cannot 
be  equal  unless  the  points  r  and  t  at  the 
junction  of  the  two  liquids  are  at  the  same 
level.  All  points  in  the  surface  of  demarca- 
tion are  therefore  in  the  same  horizontal 
plane. 

The  same  reasoning  can  be  extended  downwards  to  any  number  of 
liquids  of  unequal  densities,  which  rest  one  upon  another,  and  shows 
that  all  the  surfaces  of  demarcation  between  them  must  be  horizontal. 


Fig.  55.— Upward  Pressure. 


LIQUIDS   IN   SUPERPOSITION. 


89 


ig.  ST. 
Four  Ele 


An  experiment  in  illustration  of  this  result  is  represented  in  Fig. 
57.  Mercury,  water,  and  oil  are  poured  into  a  glass  jar.  The 
mercury,  being  the  heaviest,  goes  to 
the  bottom;  the  oil,  being  the  lightest, 
floats  at  the  top;  and  the  surfaces  of 
contact  of  the  liquids  are  seen  to  be 
horizontal. 

Even  when  liquids  are  employed  which 
gradually  mix  with  one  another,  as 
water  and  alcohol,  or  fresh  water  and 
salt  water,  so  that  there  is  no  definite 
surface  of  demarcation,  but  a  gradual 
increase  of  density  with  depth,  it  still 
remains  true  that  the  density  at  all 
points  in  a  horizontal  plane  is  the  same. 

146.  Two   Liquids    in    Bent    Tube. — 
If  we  pour  mercury  into  a  bent   tube 

open  at  both  ends  (Fig.  58),  and  then  pour  water  into  one  of 
the  arms,  the  heights  of  the  two  liquids  above  the  surface  of  junction 
will  be  very  unequal, 
as  shown  in  the  figure. 
The  general  rule  for  the 
equilibrium  of  any  two 
liquids  in  these  circum- 
•  stances  is  that  their 
heights  above  the  surface 
of  junction  must  be  in- 
versely as  their  densities, 
since  they  correspond  to 
equal  pressures. 

147.  Experiment    of 
Pascal's  Vases. — Since 
the  amount  of  pressure  < 
on  a  horizontal  area  A  ] 
at  the  depth  h  in  a  liquid 
is  whA.,  where  w  denotes 
the  weight  of  unit  volume 
of  the  liquid,  it  follows 

that  the  pressure  on  the  bottom  of  a  vessel  containing  liquid  is  not 
affected   by  the  breadth  or  narrowness  of   the  upper  part  of   the 


Fig.  5S.— Equilibrium  of  Two  Fluids  in  Communicating 
Vessels. 


90 


HYDROSTATICS. 


vessel,  provided  the  height  of  the  free  surface  of  the  liquid  be  given. 
Pascal  verified  this  fact  by  an  experiment  which  is  frequently  ex- 
hibited in  courses  of  physics.  The  apparatus  employed  (Fig.  59)  is 
a  tripod  supporting  a  ring,  into  which  can  be  screwed  three  vessels 
of  different  shapes,  one  widened  upwards,  another  cylindrical,  and 
the  third  tapering  upwards.  Beneath  the  ring  is  a  movable  disc 


Fig.  59.— Experiment  of  Pascal's  Vases. 

supported  by  a  string  attached  to  one  of  the  scales  of  a  balance. 
Weights  are  placed  in  the  other  scale  in  order  to  keep  the  disc 
pressed  against  the  ring.  Let  the  cylindrical  vase  be  mounted  on 
the  tripod,  and  filled  up  with  water  to  such  a  level  that  the  pressure 
is  just  sufficient  to  detach  the  disc  from  the  ring.  An  indicator, 
shown  in  the  figure,  is  used  to  mark  the  level  at  which  this  takes 
place.  Let  the  experiment  be  now  repeated  with  the  two  other 
vases,  and  the  disc  will  be  detached  when  the  water  has  reached  the 
same  level  as  before. 

In  the  case  of  the  cylindrical  vessel,  the  pressure  on  the  bottom 
is  evidently  equal  to  the  weight  of  the  liquid.     Hence  in  all  three 


PRESSUEE   ON   VESSEL. 


91 


Fig.  60.-Total  Pressure. 


cases  the  pressure  on  the  bottom  of  the  vessel  is  equal  to  the  weight 
of  a  cylindrical  column  of  the  liquid,  having  the  bottom  as  its  base, 
and  having  the  same  height  as  the  liquid  in  the  vessel. 

148.  Resultant  Pressure  on  Vessel. — The  pressure  exerted  by  the 
bottom  of  the  vessel  upon  the  stand  on  which  it  rests,  consists  of  the 
weight  of  the  vessel  itself,  together  with  the  resultant  pressure  of 
the  contained  liquid  against  it.  The  actual  pressure  of  the  liquid 
against  any  portion  of  the  vessel  is  normal 
to  this  portion,  and  if  we  resolve  it  into  two 
components,  one  vertical  and  the  other  hori- 
zontal, only  the  vertical  component  need  be 
attended  to,  in  computing  the  resultant; 
for  the  horizontal  components  will  always 
destroy  one  another.  At  such  points  as 
n,  ri  (Fig.  60)  the  vertical  component  is 
downwards;  at  s  and  s'  it  is  upwards;  at 
r  and  r  there  is  no  vertical  component; 
and  at  AB  the  whole  pressure  is  vertical. 
It  can  be  demonstrated  mathematically  that 

the  .resultant  pressure  is  always  equal  to  the  total  weight  of  the 
contained  liquid;  a  conclusion  which  can  also  be  deduced  from  the 
consideration  that  the  pressure  exerted  by  the  vessel  upon  the  stand 
on  which  it  rests  must  be  equal  to  its  own  weight  together  with 
that  of  its  contents. 

Some  cases  in  which  the  proof  above  indicated  becomes  especially 
obvious,  are  represented  in 
Fig.  61.  In  the  cylindrical 
vessel  ABDC,  it  is  evident 
that  the  only  pressure  trans- 
mitted to  the  stand  is  that 
exerted  upon  the  bottom, 
which  is  equal  to  the  weight 
of  the  liquid.  In  the  case 
of  the  vessel  which  is  wider 
at  the  top,  the  stand  is  subjected  to  the  weight  of  the  liquid  column 
ABSK,  which  presses  on  the  bottom  AB,  together  with  the  columns 
GHKC,  RLDS,  pressing  on  GH  and  RL;  the  sum  of  which  weights 
composes  the  total  weight  of  liquid  contained  in  the  vessel.  Finally, 
in  the  third  case,  the  pressure  on  the  bottom  AB,  which  is  equal  to 
the  weight  of  a  liquid  column  ABSK,  must  be  diminished  by  the 


A  B          A 

Fig.  61.—  Hydrostatic  Paradox. 


92 


HYDROSTATICS. 


upward  pressures  on  HG  and  KL.  These  last  being  represented  by 
liquid  columns  HGCK,  RLSD,  there  is  only  left  to  be  transmitted  to 
the  stand  a  pressure  equal  to  the  weight  of  the  water  in  the  vessel. 
-  149.  Back  Pressure  in  Discharging  Vessel. — The  same  analysis 
which  shows  that  the  resultant  vertical  pressure  of  a  liquid  against 
the  containing  vessel  is  equal  to  the  weight  of  the  liquid,  shows  also 
that  the  horizontal  components  of  the  pressures  destroy  one  another. 
This  conclusion  is  in  accordance  with  everyday  experience.  How- 
ever susceptible  a  vessel  may  be  of  horizontal  displacement,  it  is 
not  found  to  acquire  any  tendency  to  horizontal  motion  by  being 
filled  with  a  liquid. 

When  a  system  of  forces  are  in  equilibrium,  the  removal  of  one 
of  them  destroys  the  equilibrium,  and  causes  the  resultant  of  the 
system  to  be  a  force  equal  and  opposite  to  the  force  removed. 
Accordingly  if  we  remove  an  element  of  one  side  of  the  containing 
vessel,  leaving  a  hole  through  which  the  liquid  can  flow  out,  the 
remaining  pressure  against  this  side  will  be  insufiicient  to  preserve 
equilibrium,  and  there  will  be  an  excess  of  pressure  in  the  opposite 
direction. 

This  conclusion  can  be  directly  verified  by  the  experiment  repre- 
sented in  Fig.  62.  A  tall  floating 
vessel  of  water  is  fitted  with  a  hori- 
zontal discharge-pipe  on  one  side  near 
its  base.  The  vessel  is  to  be  filled 
with  water,  and  the  discharge-pipe 
opened  while  the  vessel  is  at  rest.  As 
the  water  flows  out,  the  vessel  will  be 
observed  to  acquire  a  velocity,  .at  first 
very  slow,  but  continually  increasing, 
in  the  opposite  direction  to  that  of 
the  issuing  stream. 

This  experiment  may  also  be  re- 
garded as  an  illustration  of  the  law 
of  action  and  reaction,  which  asserts 
that  momentum  cannot  be  imparted 
to  any  body  without  equal  and  opposite  momentum  being  imparted 
to  some  other  body.  The  water  in  escaping  from  the  vessel 
acquires  horizontal  momentum  in  one  direction,  and  the  vessel  with 
its  remaining  contents  acquires  horizontal  momentum  in  the  opposite 
direction. 


Fig.  62. -Backward  Movement  of 
Discharging  Vessel. 


BACKWARD   MOVEMENT. 


93 


The  movements  of  the  vessel  in  this  experiment  are  slow.  More 
marked  effects  of  the  same  kind  can  be  obtained  by  means  of  the 
hydraulic  tourn- 
iquet (Fig.  63), 
which  when  made 
on  a  larger  scale 
is  called  Barker's 
mill.  It  consists  of 
a  vessel  of  water 
free  to  rotate 
about  a  vertical 
axis,  and  having 
at  its  lower  end 
bent  arms  through 
which  the  water 
is  discharged  hori-  4 
zontally,  thef| 
direction  of  dis-  jj 
charge  beingS 
nearly  at  right 
angles  to  a  line 
joining  the  dis- 
charging orifice  to 
the  axis.  The  unbalanced  pressures  at  the  bends  of  the  tube, 
opposite  to  the  openings,  cause  the  apparatus  to  revolve  in  the 
opposite  direction  to  the  issuing  liquid. 

150.  Total  and  Resultant  Pressures.  Centre  of  Pressure. — The 
intensity  of  pressure  on  an  area  which  is  not  horizontal  is  greatest 
on  those  parts  which  are  deepest,  and  the  average  intensity  can  be 
shown  to  be  equal  to  the  actual  intensity  at  the  centre  of  gravity 
of  the  area.  Hence  if  A  denote  the  area,  h  the  depth  of  its  centre 
of  gravity,  and  w  the  weight  of  unit  volume  of  the  liquid,  the  total 
pressure  will  be  w  Ah.  Strictly  speaking,  this  is  the  pressure  due 
to  the  weight  of  the  liquid,  the  transmitted  atmospheric  pressure 
being  left  out  of  aceount. 

In  attaching  numerical  values  to  w,  A,  and  h,  the  same  unit  of 
length  must  be  used  throughout.  For  example,  if  h  be  expressed 
in  feet,  A  must  be  expressed  in  square  feet,  and  w  must  stand  for 
the  weight  of  a  cubic  foot  of  the  liquid. 

When  we  employ  the  centimetre  as  the  unit  of  length,  the  value 


Fig.  63.— Hydraulic  Tourniquet. 


HYDROSTATICS. 


of  w  will  be  sensibly  I  gramme  if  the  liquid  be  water,  so  that  the 
amount  of  pressure  in  grammes  will  be  simply  the  product  of  the 
depth  of  the  centre  of  gravity  in  centimetres  by  the  area  in  square 
centimetres.  For  any  other  liquid,  the  pressure  will  be  found  by 
multiplying  this  product  by  the  specific  gravity  of  the  liquid. 

These  rules  for  computing  total  pressure  hold  for  areas  of  all 
forms,  whether  plane  or  curved;  but  the  investigation  of  the  total 
pressure  on  an  area  which  is  not  plane  is  a  mere  mathematical 
exercise  of  no  practical  importance;  for  as  the  elementary  pressures 
in  this  case  are  not  parallel,  their  sum  (which  is  the  total  pressure) 
is  not  the  same  thing  as  their  resultant. 

For  a  plane  area,  in  whatever  position,  the  elementary  pressures, 
being  everywhere  normal  to  its  plane,  are  parallel  and  give  a  resul- 
tant equal  to  their  sum;  and  it  is  often  a  matter  of  interest  to 
determine  that  point  in  the  area  through  which  the  resultant  passes. 
This  point  is  called  the  Centre  of  Pressure.  It  is  not  coincident 
with  the  centre  of  gravity  of  the  area  unless  the  pressure  be  of 
equal  intensity  over  the  whole  area.  When  the  area  is  not  hori- 
zontal, the  pressure  is  most  intense  at  those  parts  of  it  which  are 
deepest,  and  the  centre  of  pressure  is  accordingly  lower  down  than 
the  centre  of  gravity.  For  a  horizontal  area  the  two  centres  are 
coincident,  and  they  are  also  sensibly  coincident  for  any  plane  area 
whose  dimensions  are  very  small  in  comparison  with  its  depth  in 
the  liquid,  for  the  pressure  over  such  an  area  is  sensibly  uniform. 

151.  Construction  for  Centre  of  Pressure. — If  at  every  point  of  a 
plane  area  immersed  in  a  liquid,  a  normal  be  drawn,  equal  to  the 
depth  of  the  point,  the  normals  will  represent  the  intensity  of 
pressure  at  the  respective  points,  and  the  volume  of  the  solid  con- 
stituted by  all  the  normals  will  represent  the  total  pressure.  That 
normal  which  passes  through  the  centre 
of  gravity  of  this  solid  will  be  the  line 
of  action  of  the  resultant,  and  will  there- 
fore pass  through  the  centre  of  pressure. 
Thus,  if  RB  (Fig.  64)  be  a  rectangular 
surface  (which  we  may  suppose  to  be 
the  surface  of  a  flood-gate  or  of  the  side 

Fig.  64,-Centre  of  Pressure.  °f  &   ^™)>  its  loW6r  side  B  being  at    the 

bottom  of  the  water  and  its  upper  side 

K  at  the  top,  the  pressure  is  zero  at  R  and  goes  on  increasing  uni- 
formly to  B.  The  normals  B6,  Dd,  Hfc,  LI,  equal  to  the  depths  of  a 


CENTRE   OF   PRESSURE. 


95 


series  of  points  in  the  line  BR  will  have  their  extremities  b,  d,  h,  I, 
in  one  straight  line.  To  find  the  centre  of  pressure,  we  must  find 
the  centre  of  gravity  of  the  triangle  RB6  and  draw  a  normal  through 
it.  As  the  centre  of  gravity  of  a  triangle  is  at  one-third  of  its 
height,  the  centre  of  pressure  will  be  at  one-third  of  the  height  of 
BR.  It  will  lie  on  the  line  joining  the  middle  points  of  the  upper 
and  lower  sides  of  the  rectangle,  and  will  be  at  one-third  of  the 
length  of  this  line  from  its  lower  end. 

The  total  pressure  will  be  equal  to  the  weight  of  a  quantity  of 
the  liquid  whose  volume  is  equal  to  that  of  the  triangular  prism 
constituted  by  the  aggregate  of  the  normals,  of  which  prism  the 
triangle  RB6  is  a  right  section.  It  is  not  difficult  to  show  that  the 
volume  of  this  prism  is  equal  to  the  product  of  the  area  of  the 
rectangle  by  the  depth  of  the  centre  of  gravity  of  the  rectangle,  in 
accordance  with  the  rule  above  given. 

152.  Whirling  Vessel.  D'Alembert's  Principle. — If  an  open  vessel 
of  liquid  is  rapidly  rotated  round  a  vertical  axis,  the  surface  of  the 
liquid  assumes  a  concave  form,  as  represented  in 
Fig.  65,  where  the  dotted  line  is  the  axis  of  rota- 
tion. When  the  rotation  has  been  going  on  at  a 
uniform  rate  for  a  sufficient  time,  the  liquid  mass 
rotates  bodily  as  if  its  particles  were  rigidly 
connected  together,  and  when  this  state  of  things 
has  been  attained  the  form  of  the  surface  is  that 
of  a  paraboloid  of  revolution,  so  that  the  section 
represented  in  the  figure  is  a  parabola. 

We  have  seen  in  §  101  that  a  particle  moving 
uniformly  in  a  circle  is  acted  on  by  a  force  directed 
towards  the  centre.  In  the  present  case,  therefore, 
there  must  be  a  force  acting  upon  each  particle  of 
the  liquid  urging  it  towards  the  axis.  This  force 
is  supplied  by  the  pressure  of  the  liquid,  which 
follows  the  usual  law  of  increase  with  depth  for  all 
points  in  the  same  vertical.  If  we  draw  a  horizon-  Fig  65  _R0tating  vessel 
tal  plane  in  the  liquid,  the  pressure  at  each  point  of  of  L'ilud- 

it  is  that  due  to  the  height  of  the  point  of  the  surface  vertically  over 
it.  The  pressure  is  therefore  least  at  the  point  where  the  plane  is  cut 
by  the  axis,  and  increases  as  we  recede  from  this  centre.  Consequently 
each  particle  of  liquid  receives  unequal  pressures  on  two  opposite 
sides,  being  more  strongly  pressed  towards  the  axis  than  from  it. 


96  HYDROSTATICS. 

Another  mode  of  discussing  the  case,  is  to  treat  it  as  one  of 
statical  equilibrium  under  the  joint  action  of  gravity  and  a  fictitious 
force  called  centrifugal  force,  the  latter  force  being,  for  each  par- 
ticle, equal  and  opposite  to  that  which  would  produce  the  actual 
acceleration  of  the  particle.  This  so-called  centrifugal  force  is 
therefore  to  be  regarded  as  a  force  directed  radially  outwards  from 
the  axis;  and  by  compounding  the  centrifugal  force  of  each  particle 
with  its  weight  we  shall  obtain  what  we  are  to  treat  as  the  resul- 
tant force  on  that  particle.  The  form  of  the  surface  will  then  be 
determined  by  the  condition  that  at  every  point  of  the  surface  the 
normal  must  coincide  with  this  resultant  force;  just  as  in  a  liquid 
at  rest,  the  normals  must  coincide  with  the  direction  of  gravity. 

The  plan  here  adopted  of  introducing  fictitious  forces  equal  and 
opposite  to  those  which  if  directly  applied  to  each  particle  of  a 
system  would  produce  the  actual  accelerations,  and  then  applying 
the  conditions  of  statical  equilibrium,  is  one  of  very  frequent  appli- 
cation, and  will  always  lead  to  correct  results.  This  principle  was 
first  introduced,  or  at  least  systematically  expounded,  by  D'Alem- 
bert,  and  is  known  as  D'Alembert's  Principle. 


CHAPTER    XIII. 


PRINCIPLE   OF  ARCHIMEDES. 


153.  Pressure  of  Liquids  on  Bodies  Immersed. — When  a  body  is 
immersed  in  a  liquid,  the  different  points  of  its  surface  are  sub- 
jected to  pressures  which  obey  the  rules  laid  down  in  the  preceding 
chapter.  As  these  pressures  increase  with  the  depth,  those  which 
tend  to  raise  the  body  exceed  those  which  tend  to  sink  it,  so  that 
the  resultant  effect  is  a  force  in  the  direction  opposite  to  that  of 
gravity. 

By  resolving  the  pressure  on  each  element  into  horizontal  and 
vertical  components,  it  can  be  shown  that  this  resultant  upward 
force  is  exactly  equal  to  the  weight  of  the  liquid  displaced  by  the 
body. 

The  reasoning  is  particularly  simple  in  the  case  of  a  right  cylinder 
(Fig.  66)  plunged  vertically  in  a  liquid.  It  is  evident,  in  the 
first  place,  that  if  we  consider  any  point  on  the 
sides  of  the  cylinder,  the  normal  pressure  on 
that  point  is  horizontal  and  is  destroyed  by  the 
equal  and  contrary  pressure  at  the  point  dia- 
metrically opposite;  hence,  the  horizontal  pres- 
sures destroy  each  other.  As  regards  the 
vertical  pressures  on  the  ends,  one  of  them, 
that  on  the  upper  end  AB,  is  in  a  downward 
direction,  and  equal  to  the  weight  of  the  liquid 
column  ABNN;  the  other,  that  on  the  lower  end  CD,  is  in  an 
upward  direction,  and  equal  to  the  weight  of  the  liquid  column 
CNND ;  this  latter  pressure  exceeds  the  former  by  the  weight  of  the 
liquid  cylinder  ABDC,  so  that  the  resultant  effect  of  the  pressure 
is  to  raise  the  body  with  a  force  equal  to  the  weight  of  the  liquid 
displaced. 


Fig.  66.-Principlo  of 
Archimedes. 


98  PRINCIPLE    OF  ARCHIMEDES. 

By  a  synthetic  process  of  reasoning,  we  may,  without  having 
recourse  to  the  analysis  of  the  different  pressures,  show  that  this 
conclusion  is  perfectly  general.  Suppose  we  have  a  liquid  mass  in 
equilibrium,  and  that  we  consider  specially  the  portion  M  (Fig.  G7); 
this  portion  is  likewise  in  equilibrium.  If  we 
suppose  it  to  become  solid,  without  any  change 
in  its  weight  or  volume,  equilibrium  will  still 
subsist.  Now  this  is  a  heavy  mass,  and  as  it 
does  not  fall,  we  must  conclude  that  the  effect 
of  the  pressures  on  its  surface  is  to  produce 
a  resultant  upward  pressure  exactly  equal  to 


Fig.  67.— Principle  of         its  weight,  and  acting  in  a  line  which  passes 

Archimedes.  .  ,,  .,  TI? 

through  its  centre  of  gravity.  It  we  now 
suppose  M  replaced  by  a  body  exactly  occupying  its  place,  the 
exterior  pressures  will  remain  the  same,  and  their  resultant  effect 
will  therefore  be  the  same. 

The  name  centre  of  buoyancy  is  given  to  the  centre  of  gravity 'of 
the  liquid  displaced, — that  is,  if  the  liquid  be  uniform,  to  the  centre 
of  gravity  of  the  space  occupied  by  the  immersed  body;  and  the 
above  reasoning  shows  that  the  resultant  pressure  acts  vertically 
upwards  in  a  line  which  passes  through  this  point.  The  results  of 
the  above  explanations  may  thus  be  included  in  the  following  pro- 
position: Every  body  immersed  in  a  liquid  is  subjected  to  a  resul- 
tant pressure  equal  to  the  weight  of  the  liquid  displaced,  and  acting 
vertically  upwards  through  the  centre  of  buoyancy. 

This  proposition  constitutes  the  celebrated  principle  of  Archimedes. 
The  first  part  of  it  is  often  enunciated  in  the  following  form :  Every 
body  immersed  in  a  liquid  loses  a  portion  of  its  weight  equal  to  the 
weight  of  the  liquid  displaced;  for  when  a  body  is  immersed  in  a 
liquid,  the  force  required  to  sustain  it  will  evidently  be  diminished 
by  a  quantity  equal  to  the  upward  pressure. 

154.  Experimental  Demonstration  of  the  Principle  of  Archimedes. — 
The  following  experimental  demonstration  of  the  principle  of  Archi- 
medes is  commonly  exhibited  in  courses  of  physics : — 

From  one  of  the  scales  of  a  hydrostatic  balance  (Fig.  68)  is  sus- 
pended a  hollow  cylinder  of  brass,  and  below  this  a  solid  cylinder, 
whose  volume  is  equal  to  the  interior  volume  of  the  hollow  cylinder; 
these  are  balanced  by  weights  in  the  other  scale.  A  vessel  of  water 
is  then  placed  below  the  cylinders,  in  such  a  position  that  the  lower 
cylinder  shall  be  immersed  in  it.  The  equilibrium  is  immediately 


EXPERIMENTAL   PKOOF. 


99 


destroyed,  and  the  upward  pressure  of  the  water  causes  the  scale 
with  the  weights  to  descend.  If  we  now  pour  water  into  the  hollow 
cylinder,  equilibrium  will  gradually  be  re-established;  and  the  beam 


Fig.  CS. — Experimental  Verification  of  Principle  of  Archimedes. 

will  be  observed  to  resume  its  horizontal  position  when  the  hollow 
cylinder  is  full  of  water,  the  other  cylinder  being  at  the  same  time 
completely  immersed.  The  upward  pressure  upon  this  latter  is  thus 
equal  to  the  weight  of  the  water  added,  that  is,  to  the  weight  of  the 
liquid  displaced. 

155.  Body  Immersed  in  a  Liquid. — It  follows  from  the  principle  of 
Archimedes  that  when  a  body  is  immersed  in  a  liquid,  it  is  subjected 
to  two  forces:  one  equal  to  its  weight  and  applied  at  its  centre  of 
gravity,  tending  to  make  the  body  descend;  the  other  equal  to  the 
weight  of  the  displaced  liquid,  applied  at  the  centre  of  buoyancy,  and 
tending  to  make  it  rise.  There  are  thus  three  different  cases  to  be 
considered: 

(1.)  The  weight  of  the  body  may  exceed  the  weight  of  the  liquid 
displaced,  or,  in  other  words,  the  mean  density  of  the  body  may  be 


100 


PRINCIPLE   OF   ARCHIMEDES. 


greater  than  that  of  the  liquid;  in  this  case,  the  body  sinks  in  the 
liquid,  as,  for  instance,  a  piece  of  lead  dropped  into  water. 

(2.)  The  weight  of  the  body  may  be  less  than  that  of  the  liquid 
displaced;  in  this  case  the  body  will  not  remain  submerged  unless 
forcibly  held  down,  but  will  rise  partly  out  of  the  liquid,  until^  the 
weight  of  the  liquid  displaced  is  equal  to  its  own  weight.  This  is 
what  happens,  for  instance,  if  we  immerse  a  piece  of  cork  in  water 
and  leave  it  to  itself. 

(3.)  The  weight  of  the  body  may  be  equal  to  the  weight  of  the 
liquid  displaced;  in  this  case,  the  two  opposite  forces  being  equal, 
the  body  takes  a  suitable  position  and  remains  in  equilibrium. 

These  three  cases  are  exemplified  in  the  three  following  experi- 
ments (Fig.  69):— 

(1.)  An  egg  is  placed  in  a  vessel  of  water;  it  sinks  to  the  bottom 


Fig.  69.— Egg  Plunged  in  Fresh  and  Salt  Water. 

of  the  vessel,  its  mean  density  being  a  little  greater  than  that  of  the 
liquid. 

(2.)  Instead  of  fresh  water,  salt  water  is  employed;  the  egg  floats 
at  the  surface  of  the  liquid,  which  is  a  little  denser  than  it. 

(3.)  Fresh  water  is  carefully  poured  on  the  salt  water;  a  mixture 
of  the  two  liquids  takes  place  where  they  are  in  contact;  and  if  the 
egg  is  put  in  the  upper  part,  it  will  be  seen  to  descend,  and,  after  a  few 
oscillations,  remain  at  rest  at  such  a  depth  that  it  displaces  its  own 
weight  of  the  liquid.  In  speaking  of  the  liquid  displaced  in  this 
case,  we  must  imagine  each  horizontal  layer  of  liquid  surrounding 
the  egg  to  be  produced  through  the  space  which  the  egg  occupies; 
and  by  the  centre  of  buoyancy  we  must  understand  the  centre  of 


"LIQUID  DISPLACED"  DEFINED. 


101 


gravity  of  the  portion  of  liquid  which  would  thus  take  the  place 
of  the  egg.  We  may  remark  that,  in  this  position  the  egg  is  in 
stable  equilibrium;  for,  if  it  rises,  the  upward  pressure  diminish- 
ing, its  weight  tends  to  make  it  descend  again;  if,  on  the  contrary, 
it  sinks,  the  pressure  increases  and  tends  to  make  it  reascend. 

156.  Cartesian  Diver. — The  experiment  of  the  Cartesian  diver, 
which  is  described  in  old  treatises  on  physics,  shows  each  of  the 
different  cases  that  can  present  themselves  when  a  body  is  immersed. 
The  diver  (Fig.  70)  consists  of  a  hollow  ball,  at  the  bottom  of  which 
is  a  small  opening 
O;  a  little  porcelain 
figure  is  attached  to 
the  ball,  and  the 
whole  floats  upon 
water  contained  in 
a  glass  vessel,  the 
mouth  of  which  is 
closed  by  a  strip  of 
caoutchouc  or  a  blad- 
der. If  we  press 
with  the  hand  on 
the  bladder,  the  air 
is  compressed,  and 
the  pressure,  trans- 
mitted through  the 
different  horizontal 
layers,  condenses  the 
air  in  the  ball,  and 
causes  the  entrance 
of  a  portion  of  the 
liquid  by  the  open- 
ing O;  the  floating 
body  becomes  heavier,  and  in  consequence  of  this  increase  of  weight 
the  diver  descends.  When  we  cease  to  press  upon  the  bladder,  the 
pressure  becomes  what  it  was  before,  some  water  flows  out  and  the 
diver  ascends.  It  must  be  observed,  however,  that  as  the  diver 
continues  to  descend,  more  and  more  water  enters  the  ball,  in  conse- 
quence of  the  increase  of  pressure,  so  that  if  the  depth  of  the  water 
exceeded  a  certain  limit,  the  diver  would  not  be  able  to  rise  again 
from  the  bottom. 


Fig.  TO.- Cartesian  Diver. 


102  PRINCIPLE   OF   ARCHIMEDES. 

If  we  suppose  that  at  a  certain  moment  the  weight  of  the  diver 
becomes  exactly  equal  to  the  weight  of  an  equal  volume  of  the  liquid, 
there  will  be  equilibrium;  but,  unlike  the  equilibrium  in  the  experi- 
ment (3)  of  last  section,  this  will  evidently  be  unstable,  for  a  slight 
movement  either  upwards  or  downwards  will  alter  the  resultant 
force  so  as  to  produce  further  movement  in  the  same  direction.  As 
a  consequence  of  this  instability,  if  the  diver  is  sent  down  below  a 
certain  depth  he  will  not  be  able  to  rise  again. 

157.  Relative  Positions  of  the  Centre  of  Gravity  and  Centre  of 
Buoyancy. — In  order  that  a  floating  body  either  wholly  or  partially 
immersed  in  a  liquid,  may  be  in  equilibrium,  it  is  necessary  that  its 
weight  be  equal  to  the  weight  of  the  liquid  displaced. 

This  condition  is  however  not  sufficient;  we  require,  in  addition, 
that  the  action  of  the  upward  pressure  should  be  exactly  opposite 
to  that  of  the  weight;  that  is,  that  the  centre  of  gravity  and  the 
centre  of  buoyancy  be  in  the  same  vertical  line;  for  if  this  were  not 
the  case,  the  two  contrary  forces  would  compose  a  couple,  the  effect 
of  which  would  evidently  be  to  cause  the  body  to  turn. 

In  the  case  of  a  body  completely  immersed,  it  is  further  necessary 
for  stable  equilibrium  that  the  centre  of  gravity  should  be  below  the 
centre  of  buoyancy;  in  fact  we  see,  by  Fig.  71,  that  in  any  other 


Fig.  71. 

Relative  Positions  of  Centre  of  Gravity  and  Centre  of  Pressure. 

position  than  that  of  equilibrium,  the  effect  of  the  two  forces 
applied  at  the  two  points  G  and  O  would  be  to  turn  the  body,  so  as 
to  bring  the  centre  of  gravity  lower,  relatively  to  the  centre  of 
buoyancy.  But  this  is  not  the  case  when  the  body  is  only  partially 
immersed,  as  most  frequently  happens.  In  this  case  it  may  indeed 
happen  that,  with  stable  equilibrium,  the  centre  of  gravity  is  below 
the  centre  of  pressure;  but  this  is  not  necessary,  and  in  the  majority 
of  instances  is  not  the  case.  Let  Fig.  72  represent  the  lower  part 
of  a  floating  body — a  boat,  for  instance.  The  centre  of  pressure 
is  at  0,  the  centre  of  gravity  at  G,  considerably  above;  if  the  body 


STABILITY   OF   FLOATATION.  103 

is  displaced,  and  takes  the  position  shown  in  the  figure,  it  will  be 
seen  that  the  effect  of  the  two  forces  acting  at  O  and  at  G  is  to 
restore  the  body  to  its  former  position.  This  difference  from  what 
takes  place  when  the  body  is  completely  immersed,  depends  upon 
the  fact  that,  in  the  case  of  the  floating  body,  the  figure  of  the 
liquid  displaced  changes  with  the  position  of  the  body,  and  the 
centre  of  buoyancy  moves  towards  the  side  on  which  the  body  is 
more  deeply  immersed.  It  will  depend  upon  the  form  of  the  body 
whether  this  lateral  movement  of  the  centre  of  buoyancy  is  sufficient 
to  carry  it  beyond  the  vertical  through  the  centre  of  gravity.  The 
two  equal  forces  which  act  on  the  body  will  evidently  turn  it  to  or 
from  the  original  position  of  equilibrium,  according  as  the  new  centre 
of  buoyancy  lies  beyond  or  falls  short  of  this  vertical.1 
-  158.  Advantage  of  Lowering  the  Centre  of  Gravity. — Although 
stable  equilibrium  may  subsist  with  the  centre  of  gravity  above  the 
centre  of  buoyancy,  yet  for  a  body  of  given  external  form  the 
stability  is  always  increased  by  lowering  the  centre  of  gravity;  as 
we  thus  lengthen  the  arm  of  the  couple  which  tends  to  right  the 
body  when  displaced.  It  is  on  this 
principle  that  the  use  of  ballast 
depends. 

159.  Phenomena  in  Apparent 
Contradiction  to  the  Principle  of 
Archimedes. — The  principle  of 
Archimedes  seems  at  first  sight  to 
be  contradicted  by  some  well- 
known  facts.  Thus,  for  instance,  if 

small  needles  are  placed  carefully       Fig-  73  _steel  NeedlM  Floath)g  on 
on  the  surface  of  water,  they  will 
remain  there  in  equilibrium  (Fig.  73).     It  is  on  a  similar  principle 

1  If  a  vertical  through  the  new  centre  of  buoyancy  be  drawn  upwards  to  meet  that 
line  in  the  body  which  in  the  position  of  equilibrium  was  a  vertical  through  the  centre 
of  gravity,  the  point  of  intersection  is  called  the  metacentre.  Evidently  when  the  forces 
tend  to  restore  the  body  to  the  position  of  equilibrium,  the  metacentre  is  above  the  centre 
of  gravity ;  when  they  tend  to  increase  the  displacement,  it  is  below.  In  ships  the  dis- 
tance between  these  two  points  is  usually  nearly  the  same  for  all  amounts  of  heeling,  and 
this  distance  is  a  measure  of  the  stability  of  the  ship. 

We  have  denned  the  metacentre  as  the  intersection  of  two  lines.  When  these  lines 
lie  in  different  planes,  and  do  not  intersect  each  other,  there  is  no  metacentre.  This 
indeed  is  the  case  for  most  of  the  displacements  to  which  a  floating  body  of  irregular 
shape  can  be  subjected.  There  are  in  general  only  two  directions  of  heeling  to  which 
metacentres  correspond,  and  these  two  directions  are  at  right  angles  to  each  other. 


104  PRINCIPLE   OF  ARCHIMEDES. 

that  several  insects  walk  on  water  (Fig.  74),  and  that  a  great 
number  of  bodies  of  various  natures,  provided  they  be  very  minute, 

can,  if  we  may  so  say,  be  placed 
on  the  surface  of  a  liquid  with- 
out penetrating  into  its  interior. 
These  curious  facts  depend  on  the 
circumstance  that  the  small  bodies 
Fig.  74. -insect  walking  on  water.  in  question  are  not  wetted  by  the 
liquid,  and  hence,  in  virtue  of 

principles  which  will  be  explained  in  connection  with  capillarity 
(Chap,  xvi.),  depressions  are  formed  around  them  on  the  liquid 
surface,  as  represented  in  Fig.  75.  The  curvature  of  the  liquid 
surface  in  the  neighbourhood  of  the  body  is  very  distinctly  shown 
by  observing  the  shadow  cast  by  the  floating  body,  when  it  is 
illumined  by  the  sun;  it  is  seen  to  be  bordered  by  luminous  bands, 
which  are  owing  to  the  refraction  of  the  rays  of  light  in  the  portion 
of  "the  liquid  bounded  by  a  curved  surface. 

The  existence  of  the  depression  about  the  floating  body  enables 
us  to  bring  the  condition  of  equilibrium  in  this 
special  case  under  the  general  enunciation  of  the 
principle  of  Archimedes.  Let  M  (Fig.  75)  be 
the  body,  CD  the  region  of  the  depression,  and 
AB  the  corresponding  portion  of  any  horizontal 
Fig.  75.  layer;  since  the  pressure  at  each  point  of  AB 

must  be  the  same  as  in  other  parts  of  the  same 
horizontal  layer,  the  total  weight  above  AB  is  the  same  as  if  M 
did  not  exist  and  the  cavity  were  filled  with  the  liquid  itself. 

We  may  thus  say  in  this  case  also  that  the  weight  of  the  floating 
body  is  equal  to  the  weight  of  the  liquid  displaced,  understanding 
by  these  words  the  liquid  which  would  occupy  the  whole  of  the 
depression  due  to  the  presence  of  the  body. 


CHAPTER   XIV. 


DENSITY  AND   ITS   DETERMINATION. 


160.  Definitions.  —  By  the  absolute  density  of  a  substance  is  meant 
the  mass  of  unit  volume  of  it.  By  the  relative  density  is  meant  the 
ratio  of  its  absolute  density  to  that  of  some  standard  substance,  or, 
what  amounts  to  the  same  thing,  the  ratio  of  the  mass  of  any  volume 
of  the  substance  in  question  to  the  mass  of  an  equal  volume  of  the 
standard  substance.  Since  equal  masses  gravitate  equally,  the  com- 
parison of  masses  can  be  effected  by  weighing,  and  the  relative  den- 
sity of  a  substance  is  the  ratio  of  its  weight  to  that  of  an  equal 
volume  of  the  standard  substance.  Water  at  a  specified  tempera- 
ture and  under  atmospheric  pressure  is  usually  taken  as  the  standard 
substance,  and  the  density  of  a  substance  relative  to  water  is  usually 
called  the  specific  gravity  of  the  substance. 

Let  V  denote  the  volume  of  a  substance,  M  its  mass,  and  D  its 
absolute  density;  then  by  definition,  we  have  Mr=VD. 

If  s  denote  the  specific  gravity  of  a  substance,  and  d  the  absolute 
density  of  water  in  the  standard  condition,  then  ~D=sd  and  M= 


When  masses  are  expressed  in  Ibs.  and  volumes  in  cubic  feet,  the 
value  of  d  is  about  62-4,  since  a  cubic  foot  of  cold  water  weighs 
about  C2-4  Ibs.1 

In  the  C.G.S.  system,  the  value  of  d  is  sensibly  unity,  since  a 
cubic  centimetre  of  water,  at  a  temperature  which  is  nearly  that  of 
the  maximum  density  of  water,  weighs  exactly  a  gramme.2 

The  gramme  is  defined,  not  by  reference  to  water,  but  by  a 
standard  kilogramme  of  platinum,  which  is  preserved  in  Paris,  and 

1  In  round  numbers,  a  cubic  foot  of  water  weighs  1000  oz.,  which  is  62'5  Ibs. 
*  According  to  the  best  determination  yet  published,  the  mass  of  a  cubic  centimetre  of 
pure  water  at  4°  is  1  '00001  3,  at  3°  is  1-000004,  and  at  2°  is  "999982. 


106  DENSITY   AND   ITS   DETERMINATION. 

of  which  several  very  carefully  made  copies  are  preserved  in  other 
places.  In  the  above  statements  (as  in  all  very  accurate  statements 
of  weights),  the  weighings  are  supposed  to  be  made  in  vacuo;  for 
the  masses  of  two  bodies  are  not  accurately  proportional  to  their 
apparent  gravitations  in  air,  unless  the  two  bodies  happen  to  have 
the  same  density. 

161.  Ambiguity  of  the  word  "  Weight."— Properly  speaking,  "  the 
weight  of  a  body  "  means  the  force  with  which  the  body  gravitates 
towards  the  earth.     This  force,  as  we  have  seen,  differs  slightly 
according  to  the  place  of  observation.     If  m  denote  the  mass  of  the 
body,  and  g  the  intensity  of  gravity  at  the  place,  the  weight  of  the 
body  is  mg.     When  the  body  is  carried  from  one  place  to  another 
without  gain  or  loss  of  material,  m  will  remain  constant  and  g  will 
vary;  hence  the  weight  mg  will  vary,  and  in  the  same  ratio  as  g. 

But  the  employment  of  gravitation  units  of  force  instead  of 
absolute  units,  obscures  this  fact.  The  unit  of  measurement  varies 
in  the  same  ratio  as  the  thing  to  be  measured,  and  hence  the 
numerical  value  remains  unaltered.  A  body  weighs  the  same 
number  of  pounds  or  grammes  at  one  place  as  at  another,  because 
the  weights  of  the  pound  and  gramme  are  themselves  proportional 
to  g.  Expressed  in  absolute  units,  the  weight  of  unit  mass  is  g,  and 
the  weight  of  a  mass  m  is  mg.  The  latter  is  m  times  the  former; 
hence  when  the  weight  of  unit  mass  is  employed  as  the  unit  of 
weight,  the  same  number  m  which  denotes  the  mass  of  a  body  also 
denotes  its  weight.  What  are  usually  called  standard  weights — 
that  is,  standard  pieces  of  metal  used  for  weighing — are  really 
standards  of  mass;  and  when  the  result  of  a  weighing  is  stated  in 
terms  of  these  standards,  (as  it  usually  is,)  the  "  weight,"  as  thus 
stated,  is  really  the  mass  of  the  body  weighed.  The  standard 
"  weights  "  which  we  use  in  our  balances  are  really  standard  masses. 
In  discussions  relating  to  density,  weights  are  most  conveniently 
expressed  in  gravitation  measure,  and  hence  the  words  mass  and 
weight  can  be  used  almost  indiscriminately. 

162.  Determination  of  Density  from  Weight  and  Volume.— The 
absolute   density  of  a  substance   can   be   directly   determined   by 
weighing  a  measured  volume  of  it.     Thus  if  v  cubic  centimetres  of 
it  weigh  m  grammes,  its  density  (in  grammes  per  cubic  centimetre) 
is  *.      This   method  can  be  easily  applied   to  solids  of  regular 
geometrical  forms;  since  their  volumes  can  be  computed  from  their 


SPECIFIC    GRAVITY   BOTTLE. 


107 


linear  measurements.  It  can  also  be  applied  to  liquids,  by  employ- 
ing a  vessel  of  known  content.  The  bottle  usually  employed  for 
this  purpose  is  a  bottle  of  thin  glass  fitted  with  a  perforated  stopper, 
so  that  it  can  be  filled  and  stoppered  without  leaving  a  space  for 
air.  The  difference  between  its  weights  when  full  and  empty  is  the 
weight  of  the  liquid  which  fills  it;  and  the  quotient  of  this  by  the 
volume  occupied  (which  can  be  determined  once  for  all  by  weighing 
the  bottle  when  filled  with  water)  is  the  density  of  the  liquid. 

The  advantage  of  employing  a  perforated  stopper  is  that  it  enables 
us  to  ensure  constancy  of  volume.  If  a  wide-mouthed  flask  were 
employed,  without  a  stopper,  it  would  be  difficult  to  pronounce 
when  the  flask  was  exactly  full.  This  source  of  inaccuracy  would 
be  diminished  by  making  the  mouth  narrower:  but  when  it  is  very 
narrow,  the  filling  and  emptying  of  the  flask  are  difficult,  and  there 
is  danger  of  forcing  in  bubbles  of  air  with  the  liquid.  When  a  per- 
forated stopper  is  employed,  the  flask  is  first  filled,  then  the  stopper 
is  inserted  and  some  of  the  liquid  is  thus  forced  up  through  the 
perforation,  overflowing  at  the  top.  When  the  stopper  has  been 
pushed  home,  all  the  liquid  outside  is  carefully  wiped  off,  and  the 
liquid  which  remains  is  as  much  as  just  fills  the  stoppered  flask 
including  the  perforation  in  the  stopper. 

In  accurate  work,  the  temperature  must  be  observed,  and  due 
allowance  made  for  its  effect  upon  volume. 

163.  Specific  Gravity  Flask  for  Solids. — The  volume  and  density  of 
a  solid  body  of  irregular  shape,  or  consisting  of  a  quantity  of  small 
pieces,  can  be  de- 
termined by  put- 
ting it  into  such 
a  bottle  (Fig.  76), 
and  weighing 
the  water  which 
it  displaces.  The 
most  convenient 
way  of  doing 
this  is  to  observe 
(1)  the  weight  of 
the  solid;  (2)  the 
weight  of  the 

bottle  full  of  water;  (3)  the  weight  of  the  bottle  when  it  contains 
the  solid,  together  with  as  much  water  as  will  fill  it  up.  If  the 


Fig  76.— Specific  Gravity  Flask  for  Soli 


108 


DENSITY  AND   ITS   DETERMINATION. 


third  of  these  results  be  subtracted  from  the  sum  of  the  first  two, 
the  remainder  will  be  the  weight  of  the  water  displaced;  which, 
when  expressed  in  grammes,  is  the  same  as  the  volume  of  the  body 
expressed  in  cubic  centimetres.  The  weight  of  the  body,  divided  by 
this  remainder,  is  the  density  of  the  body. 

164.  Method  by  Weighing  in  Water.— The  methods  of  determining 
density  which  we  are  now  about  to  describe  depend  upon  the  prin- 
ciple of  Archimedes. 

One  of  the  commonest  ways  of  determining  the  density  of  a  solid 
body  is  to  weigh  it  first  in  air  and  then  in  water  (Fig.  77)  the 


Fig.  77.-Specific  Gravity  of  Solids.  Fig.  7S.-Specific  Gravity  of  Liquids. 

counterpoising  weights  being  in  air.  Since  the  loss  of  weight  due 
to  its  immersion  in  water  is  equal  to  the  weight  of  the  same  volume 
of  water,  we  have  only  to  divide  the  weight  in  air  by  this  loss  of 
weight.  We  shall  thus  obtain  the  relative  density  of  the  body 
as  compared  with  water-in  other  words,  the  specific  gravity  of  the 


WEIGHING  IN  WATER.  109 

Thus,  from  the  observations 

Weight  in  air,         125  gm. 
Weight  in  water,    100   „ 
Loss  of  weight,        25   „ 

we  deduce 

1  or 

^ ;  =  5  =  density. 

A  very  fine  and  strong  thread  or  fibre  should  be  employed  for  sus- 
pending the  body,  so  that  the  volume  of  liquid  displaced  by  this 
thread  may  be  as  small  as  possible. 

165.  Weighing  in  Water,  with  a  Sinker.— If  the  body  is  lighter 
than  water,  we  may  employ  a  sinker — that  is,  a  piece  of  some  heavy 
material  attached  to  it,  and  heavy  enough  to  make  it  sink.     It  is 
not  necessary  to  know  the  weight  of  the  sinker  in  air,  but  we  must 
observe  its  weight  in  water.     Call  this  s.     Let  w  be  the  weight  of 
the  body  in  air,  and  w  the  weight  of  the  body  and  sinker  together 
in  water.     Then  w'  will  be  less  than  s.     The  body  has  an  apparent 
upward   gravitation   in  water   equal  to   s—w',  showing   that   the 
resultant   pressure   upon   it   exceeds   its   weight   by  this   amount. 
Hence  the  weight  of  the  liquid  displaced  is  w-\- s—w, and  the  specific 

gravity  of  the  body  is  w     ™  _  ^ 

If  any  other  liquid  than  water  be  employed  in  the  methods 
described  in  this  and  the  preceding  section,  the  result  obtained  will 
be  the  relative  density  as  compared  with  that  liquid.  The  result 
must  therefore  be  multiplied  by  the  density  of  the  liquid,  in  order 
to  obtain  the  absolute  density. 

166.  Density  of  Liquid  Inferred  from  Loss  of  Weight. — The  densities 
of  liquids  are  often  determined  by  observing  the  loss  of  weight  of  a 
solid  immersed  in  them,  and  dividing  by  the  known  volume  of  the 
solid  or  by  its  loss  of  weight  in  water. 

Thus,  from  the  observations 

Weight  in  air,         200  gm 
Weight  in  liquid,    120  „ 
Weight  in  water,    110  „ 

we  deduce 

Loss  in  liquid,  80.     Loss  in  water,  90. 
Density  of  liquid,  *°  =  * 

A  glass  ball  (sometimes  weighted  with  mercury,  as  in  Fig.  78)  is 
the  solid  most  frequently  employed  for  such  observations. 


110 


DENSITY   AND   ITS   DETERMINATION. 


167.  Measurement  of  Volumes  of  Solids  by  Loss  of  Weight.— The 
volume  of  a  solid  body,  especially  if  of  irregular  shape,  can  usually 
be  determined  with  more  accuracy  by  weighing  it  in  a  liquid  than  by 
any  other  method.     If  it  weigh  w  grammes  in  air,  and  w'  grammes 
in  water,  its  volume  is  w—w  cubic  centimetres,  since  it  displaces 
w—w'  grammes  of  water.     The  mean  diameter  of  a  wire  can  be 
very  accurately  determined  by  an  observation  of   this   kind   for 
volume,   combined  with    a   direct    measurement    of    length.     The 
volume  divided  by  the  length  will  be  the  mean  sectional  area, 
which  is  equal  to  vr2,  where  r  is  the  radius. 

168.  Hydrometers. — The  name  hydrometer  is  given  to  a  class  of 
instruments  used  for  determining  the  densities  of  liquids  by  observ- 
ing either  the  depths  to  which  they  sink  in  the  liquids  or  the 


>lson's  Hydrometer. 

weights  required  to  be  attached  to  them  to  make  them  sink  to  a 
given  depth.  According  as  they  are  to  be  used  in  the  latter  or  the 
tormer  of  these  two  ways,  they  are  called  hydrometers  of  constant 
or  of  variable  immersion.  The  name  areometer  (from  Apcuoc,  rare) 
is  used  as  synonymous  with  hydrometer,  being  probably  borrowed 
om  the  French  name  of  these  instruments,  artomtore.  The  hydro- 


NICHOLSON'S  HYDROMETER.  Ill 

meters  of  constant  immersion  most  generally  known  are  those  of 
Nicholson  and  Fahrenheit. 

169.  Nicholson's  Hydrometer. — This  instrument,  which  is  repre- 
sented in  Fig.  79,  consists  of  a  hollo*v  cylinder  of  metal  with  conical 
ends,  terminated  above  by  a  very  thin  rod  bearing  a  small  dish,  and 
carrying  at  its  lower  end  a  kind  of  basket.     This  latter  is  of  such 
weight  that  when  the  instrument  is  immersed  in  water  a  weight  of 
100  grammes  must  be  placed  in  the  dish  above  in  order  to  sink  the 
apparatus  as  far  as  a  certain  mark  on  the  rod.     By  the  principle  of 
Archimedes,  the  weight  of  the  instrument,  together  with  the  100 
grammes  which  it  carries,  is  equal  to  the  weight  of  the  water  dis- 
placed.    Now,  let  the  instrument  be  placed  in  another  liquid,  and 
the  weights  in  the  dish  above  be  altered  until  they  are  just  sufficient 
to  make  the  instrument  sink  to  the  mark  on  the  rod.     If  the  weights 
in  the  dish  be  called  w,  and  the  weight  of  the  instrument  itself  W, 
the  weight  of  liquid  displaced  is  now  W  +  iv,  whereas  the  weight 
of  the  same  volume  of  water  was    W  +  100;   hence  the  specific 

gravity  of  the  liquid  is  ^loo' 

This  instrument  can  also  be  used  either  for  weighing  small  solid 
bodies  or  for  finding  their  specific  gravities.  To  find  the  weight  of 
a  body  (which  we  shall  suppose  to  weigh  less  than  100  grammes),  it 
must  be  placed  in  the  dish  at  the  top,  together  with  weights  just 
sufficient  to  make  the  instrument  sink  in  water  as  far  as  the  mark. 
Obviously  these  weights  are  the  difference  between  the  weight  of 
the  body  and  100  grammes. 

To  find  the  specific  gravity  of  a  solid,  we  first  ascertain  its 
weight  by  the  method  just  described;  we  then  transfer  it  from 
the  dish  above  to  the  basket  below,  so  that  it  shall  be  under 
water  during  the  observation,  and  observe  what  additional  weights 
must  now  be  placed  in  the  dish.  These  additional  weights 
represent  the  weight  of  the  water  displaced  by  the  solid;  and 
the  weight  of  the  solid  itself  divided  by  this  weight  is  the  specific 
gravity  required. 

170.  Fahrenheit's  Hydrometer. — This  instrument,  which  is  repre- 
sented in  Fig.  80,  is  generally  constructed  of  glass,  and  differs  from 
Nicholson's  in  having  at  its  lower  extremity  a  ball  weighted  with 
mercury  instead  of  the  basket.     It  resembles  it  in  having  a  dish  at 
the  top,  in  which  weights  are  to  be  placed  sufficient  to  sink  the 
instrument  to  a  definite  mark  on  the  stem. 


112 


DENSITY  AND   ITS   DETERMINATION. 


Hydrometers  of  constant  immersion,  though  still  described  in 
text-books,  have  quite  gone  out  of  use  for  practical  work. 

171  Hydrometers  of  Variable  Immersion.— These  instruments  are 
usually  of  the  forms  represented^  A,  B,  C,  Fig.  81.  The  lower  end 
is  weighted  with  mercury  in  order  to  make  the  instrument  sink  to 
a  convenient  depth  and  preserve  an  upright  position.  The  stem  is 
cylindrical,  and  is  graduated,  the  divisions  being  frequently  marked 


Fig.  SO. — Fahrenheit's  Hydrometer. 


Fig.  SI.— Forms  of  Hydrometers. 


upon  a  piece  of  paper  inclosed  within  the  stem,  which  must  in  this 
case  be  of  glass.  It  is  evident  that  the  instrument  will  sink  the 
deeper  the  less  is  the  specific  gravity  of  the  liquid,  since  the  weight 
of  the  liquid  displaced  must  be  equal  to  that  of  the  instrument. 
Hence  if  any  uniform  system  of  graduation  be  adopted,  so  that  all 
the  instruments  give  the  same  readings  in  liquids  of  the  same  densi- 
ties, the  density  of  a  liquid  can  be  obtained  by  a  mere  immersion 
of  the  hydrometer — an  operation  not  indeed  very  precise,  but  very 
easy  of  execution.  These  instruments  have  thus  come  into  general 
use  for  commercial  purposes  and  in  the  excise. 

172.  General  Theory  of  Hydrometers  of  Variable  Immersion. — Let 
V  be  the  volume  of  a  hydrometer  which  is  immersed  when  the  in- 
strument floats  freely  in  a  liquid  whose  density  is  d,  then  Nd  repre- 


HYDROMETERS.  113 

sents  the  weight  of  liquid  displaced,  which  by  the  principle  of  Archi- 
medes is  the  same  as  the  weight  of  the  hydrometer  itself.  If  V,  d 
be  the  corresponding  values  for  another  liquid,  we  have  therefore 

Vd^V'd',  OTd:d'::V:V, 

that  is,  the  density  varies  inversely  as  the  volume  immersed.  Let 
dlt  d.2,  c£3...be  a  series  of  densities,  and  V1}  V2,  V3...the  corresponding 
volumes  immersed,  then  we  have 

111 

dlt  d2,  o3...  proportional  to  — ,  — ,  — ... 
and  Vb  Va  Vs...  proportional  to  --,  -  ,      ... 

Hence,  if  we  wish  the  divisions  to  indicate  equal  differences  of  den- 
sity, we  must  place  them  so  that  the  corresponding  volumes  im- 
mersed form  a  harmonical  progression.  This  implies  that  the  dis- 
tances between  the  divisions  must  diminish  as  the  densities  increase. 
The  following  investigation  shows  how  the  density  of  a  liquid 
may  be  computed  from  observations  made  with  a  hydrometer  gradu- 
ated with  equal  divisions.  It  is  necessary  first  to  know  the  divisions 
to  which  the  instrument  sinks  in  two  liquids  of  known  densities. 
Let  these  divisions  be  numbered  nlt  n.2,  reckoning  from  the  top 
downwards,  and  let  the  corresponding  densities  be  d1}  d.2.  Now  if 
we  take  for  our  unit  of  volume  one  of  the  equal  parts  on  the  stem, 
and  if  we  take  c  to  denote  the  volume  which  is  immersed  when  the 
instrument  sinks  to  the  division  marked  zero,  it  is  obvious  that  when 
the  instrument  sinks  to  the  Tith  division  (reckoned  downwards  on 
the  stem  from  zero)  the  volume  immersed  is  c—n,  and  if  the  corre- 
sponding density  be  called  d,  then  (c  —  n)  d  is  the  weight  of  the 
hydrometer.  We  have  therefore 

(c-Wi)  di  =  (c-ni)  d-2,  whence  c  =  "'f|1  ~  ^-— . 


This  value  of  c  can  be  computed  once  for  all. 
Then  the  density  D  corresponding  to  any  other  division  N  can  be 
found  from  the  equation 

(c  -  N)  D  =  (c  -  n,)  rft  which  gives  D  =  — "X- 

•  173.  Beaume's  Hydrometers. — In  these  instruments  the  divisions 
are  equidistant.  There  are  two  distinct  modes  of  graduation,  accord- 
ing as  the  instrument  is  to  be  used  for  determining  densities  greater 
or  less  than  that  of  water.  In  the  former  case  the  instrument  is 
8 


114 


DENSITY   AND   ITS   DETERMINATION. 


called  a  salimeter,  and  is  so  constructed  that  when  immersed  in  pure 
water  of  the  temperature  12°  Cent,  it  sinks  nearly  to  the  top  of  the 
stem,  and  the  point  thus  determined  is  the  zero  of  the  scale.  It  is 
then  immersed  in  a  solution  of  15  parts  of  salt  to  85  of 
water,  the  density  of  which  is  about  1116,  and  the  point 
to  which  it  sinks  is  marked  15.  The  interval  is  divided 
into  15  equal  parts,  and  the  graduation  is  continued  to 
the  bottom  of  the  stem,  the  length  of  which  varies  accord 
ing  to  circumstances;  it  generally  terminates  at  the 
degree  66,  which  corresponds  to  sulphuric  acid,  whose 
density  is  commonly  the  greatest  that  it  is  required 
to  determine.  Referring  to  the  formulae  of  last  section, 
we  have  here 


whence 


15x1-116 


When  the  instrument  is  intended  for  liquids  lighter  than  water,  it 
is  called  an  alcoholimeter.  In  this  case  the  point  to  which  it  sinks 
in  water  is  near  the  bottom  of  the  stem,  and  is 
marked  10;  the  zero  of  the  scale  is  the  point  to 
which  it  sinks  in  a  solution  of  10  parts  of  salt  to 
90  of  water,  the  density  of  which  is  about  TOSS, 
the  divisions  in  this  case  being  numbered  upward 
from  zero. 

In  order  to  adapt  the  formulae  of  last  section  to 
the  case  of  graduations  numbered  upwards,  it  is 
merely  necessary  to  reverse  the  signs  of  %,  n.2,  and 
N;  that  is  we  must  put 


Fig.  83.      Fig.... 

Baume's    Alcoholi- 

meters. 


and  as  we  have  now  7^ =10, 
the  formulae  give1 

10  128 


=1,  n.2=0,  (72=  1*08 


174.  Twaddell's  Hydrometer. — In  this  instrument  the  divisions  are 

1  On  comparing  the  two  formulae  for  D  in  this  section  with  the  tables  in  the  Appendix 
to  Miller's  Chemical  Physics,  I  find  that  as  regards  the  salimeter  they  agree  to  two  places 
of  decimals  and  very  nearly  to  three.  As  regards  the  alcoholimeter,  the  table  in  Miller 
implies  that  e  is  about  136,  which  would  make  the  density  corresponding  to  the  zero  of 
the  scale  about  T074. 


DENSITY   OF   MIXTURES.  115 

placed  not  as  in  Beaume's,  at  equal  distances,  but  at  distances 
corresponding  to  equal  differences  of  density.  In  fact  the  specific 
gravity  of  a  liquid  is  found  by  multiplying  the  reading  by  5,  cutting 
off  three  decimal  places,  and  prefixing  unity.  Thus  the  degree  1 
indicates  specific  gravity  T005,  2  indicates  1O10,  &c. 

175.  Gay-Lussac's  Centesimal  Alcoholimeter. — When  a  hydrometer 
is  to  be  used  for  a  special  purpose,  it  may  be  convenient  to  adopt  a 
mode  of  graduation  different  in  principle  from  any  that  we  have 
described  above,  and  adapted  to  give  a  direct  indication  of 

the  proportion  in  which  two  ingredients  are  mixed  in  the 
fluid  to  be  examined.  It  may  indicate,  for  example,  the 
quantity  of  salt  in  sea- water,  or  the  quantity  of  alcohol  in  a 
spirit  consisting  of  alcohol  and  water.  Where  there  are 
three  or  more  ingredients  of  different  specific  gravities 
the  method  fails.  Gay-Lussac's  alcoholimeter  is  graduated 
to  indicate,  at  the  temperature  of  15°  Cent.,  the  percentage 
of  pure  alcohol  in  a  specimen  of  spirit.  At  the  top  of  the 
stem  is  100,  the  point  to  which  the  instrument  sinks  in 
pure  alcohol,  and  at  the  bottom  is  0,  to  which  it  sinks  in 
water.  The  position  of  the  intermediate  degrees  must  be 
determined  empirically,  by  placing  the  instrument  in  mix- 
tures of  alcohol  and  water  in  known  proportions,  at  the  Fig  ss. 
temperature  of  15°.  The  law  of  density,  as  depending  on  Aicohoii- 
the  proportion  of  alcohol  present,  is  complicated  by  the  fact 
that,  when  alcohol  is  mixed  with  water,  a  diminution  of  volume 
(accompanied  by  rise  of  temperature)  takes  place. 

176.  Specific  Gravity  of  Mixtures. — When  two  or  more  substances 
are  mixed  without  either  shrinkage  or  expansion  (that  is,  when  the 
volume  of  the  mixture  is  equal  to  the  sum  of  the  volumes  of  the 
components),  the  density  of  the  mixture  can  easily  be  expressed  in 
terms  of  the  quantities  and  densities  of  the  components. 

First,  let  the  volumes  vlf  v2,  v3  .  .  .  of  the  components  be  given, 
together  with  their  densities  d1,  d2,  d3  .  .  . 
Then  their  masses  (or  weights)  are  v^d^  v2d.2,  V3d3  .  .  . 
The  mass  of  the  mixture  is  the  sum  of  these  masses,  and  its  volume 
is  the  sum  of  the  volumes  vlt  v2,  %  .  .  .  ;  hence  its  density  is 


Secondly,  let  the  weights  or  masses  mv  mz,  mz  .  .  .  of  the  com- 
ponents be  given,  together  with  their  densities  dli  d2,  dz  .  .  . 


116 


DENSITY  AND   ITS  DETERMINATION. 


Then  their  volumes  are  ~\  ^,  J  •  •  • 

The  volume  of  the  mixture  is  the  sum  of  these  volumes,  and  its  mass 

is  w,  +  ma  +  ms  +  .  .  .  ;  hence  its  density  is 


177.  Graphical  Method  of  Graduation.  —  When  the  points  on  the 
stem  which  correspond  to  some  five  or  six  known  densities,  nearly 
equidifferent,  have  been  determined,  the  intermediate  graduations 
can  be  inserted  with  tolerable  accuracy  by  the  graphical  method  of 
interpolation,  a  method  which  has  many  applications  in  physics 
besides  that  which  we  are  now  considering.  Suppose  A  and  B 
(Fig.  86)  to  represent  the  extreme  points,  and  I,  K,  L,  R  intermediate 

points,  all  of  which  correspond 
to  known  densities.  Erect 
ordinates  (that  is  to  say,  per- 
pendiculars) at  these  points, 
proportional  to  the  respective 
densities,  or  (which  will  serve 
our  purpose  equally  well) 
erect  ordinates  II',  KK',  LL', 
RR',  BC  proportional  to  the 
excesses  of  the  densities  at 
I,  K,  L,  R,  B  above  the  den- 
sity at  A.  Any  scale  of  equal 
parts  can  be  employed  for 

laying  off  the  ordinates,  but  it  is  convenient  to  adopt  a  scale  which 
will  make  the  greatest  ordinate  BC  not  much  greater  nor  much 
less  than  the  base-line  AB.  In  the  figure,  the  density  at  B  is 
supposed  to  be  1-80,  the  density  at  A  being  1.  The  difference 
of  density  is  therefore  '80,  as  indicated  by  the  figures  80  on  the 
scale  of  equal  parts.  Having  erected  the  ordinates,  we  must 
draw  through  their  extremities  the  curve  AI'K'L'R'C,  making  it 
as  ^free  from  sudden  bends  as  possible,  as  it  is  upon  the  regu- 
larity of  this  curve  that  the  accuracy  of  the  interpolation  depends. 
Then  to  find  the  point  on  the  stem  AB  at  which  any  other 
density  is  to  be  marked-say  1«60,  we  must  draw  through  the 
^Oth  division,  on  the  line  of  equal  parts,  a  horizontal  fine  to 
meet  the  curve,  and,  through  the  point  thus  found  on  the  curve, 


I  K  L    1.4       R  1.6        B 

Fig.  86.—  Graphical  Method  of  Graduation. 


GRAPHICAL   INTERPOLATION.  117 

draw  an  ordinate.  This  ordinate  will  meet  the  base-line  AB  in  the 
required  point,  which  is  accordingly  marked  1'6  in  the  figure.  The 
curve  also  affords  the  means  of  solving  the  converse  problem,  that 
is,  of  finding  the  density  corresponding  to  any  given  point  on  the 
stem.  At  the  given  point  in  AB,  which  represents  the  stem,  we 
must  draw  an  ordinate,  and  through  the  point  where  this  meets  the 
curve  we  must  draw  a  horizontal  line  to  meet  the  scale  of  equal 
parts.  The  point  thus  determined  on  the  scale  of  equal  parts  indi- 
cates the  density  required,  or  rather  the  excess  of  this  density  above 
the  density  of  A. 


CHAPTER    XV. 


VESSELS  IN  COMMUNICATION — LEVELS. 


178.  Liquids  tend  to  Find  their  own  Level. — When  a  liquid  is 
contained  in  vessels  communicating  with  each  other,  and  is  in 
equilibrium,  it  stands  at  the  same  height  in  the  different  parts  of 
the  system,  so  that  the  free  surfaces  all  lie  in  the  same  horizontal 
plane.  This  is  obvious  from  the  considerations  pointed  out  in 
§§  138,  139,  being  merely  a  particular  case  of  the  more  general  law 
that  points  of  a  liquid  at  rest  which  are  at  the  same  pressure  are  at 
the  same  level. 

In  the  apparatus  represented  in  Fig.  87,  the  liquid  is  seen  to  stand 

at  the  same  height  in 

|7     V\  ,^,J  the    principal    vessel 

\1        if  and  in  the  variously 

shaped  tubes  com- 
municating with  it. 
If  one  of  these  tubes  is 
cut  off  at  a  height  less 
than  that  of  the  liquid 
in  the  principal  vessel, 
and  is  made  to  termin- 
ate ina  narrowmouth, 
the  liquid  will  be  seen 
to  spout  up  nearly  to 
the  level  of  that  in 
the  principal  vessel. 


Fig.  87.— Vessels  in  Communication. 

This  tendency  of  liquids  to  find  their  own  level  is  utilized  for  the 
water-supply  of  towns.  Water  will  find  its  way  from  a  reservoir 
through  pipes  of  any  length,  provided  that  all  parts  of  them  are 
below  the  level  of  the  water  in  the  reservoir.  It  is  necessary  how- 


WATER   SUPPLY.  119 

ever  to  distinguish  between  the  conditions  of  statical  equilibrium 
and  the  conditions  of  flow.  If  no  water  were  allowed  to  escape 
from  the  pipes  in  a  town,  their  extremities  might  be  carried  to  the 
height  of  the  reservoir  and  they  would  still  be  kept  full.  But  in 
practice  there  is  a  continual  abstraction  of  energy,  partly  in  the 
shape  of  the  kinetic  energy  of  the  water  which  issues  from  taps, 
often  with  considerable  velocity,  and  partly  in  the  shape  of  work 
done  against  friction  in  the  pipes.  When  there  is  a  continual  draw- 
ing off  from  various  points  of  a  main,  the  height  to  which  the  water 
will  rise  in  the  houses  which  it  supplies  is  least  in  those  which  are 
most  distant  from  the  reservoir. 

179.  Water-level. — The  instrument  called  the  water-level  is  another 
illustration  of  the  same  principle.  It  consists  of  a  metal  tube  Ib, 
bent  at  right  angles  at  its  extremities.  These  carry  two  glass  tubes 


Fig.  88.— Water-leveL 

aa,  very  narrow  at  the  top,  and  of  the  same  diameter.  The  tube 
rests  on  a  tripod  stand,  at  the  top  of  which  is  a  joint  that  enables 
the  observer  to  turn  the  apparatus  and  set  it  in  any  direction.  The 
tube  is  placed  in  a  position  nearly  horizontal,  and  water,  generally 
coloured  a  little,  is  poured  in  until  it  stands  at  about  three-fourths  of 
the  height  of  each  of  the  glass  tubes. 

By  the  principle  of  equilibrium  in  vessels  communicating  with 
each  other,  the  surfaces  of  the  liquid  in  the  two  branches  are  in  the 
same  horizontal  plane,  so  that  if  the  line  of  the  observer's  sight  just 
grazes  the  two  surfaces  it  will  be  horizontal. 

This  is  the  principle  of  the  operation  called  levelling,  the  object  of 
which  is  to  determine  the  difference  of  vertical  height,  or  difference 
of  level,  between  two  given  points.  Suppose  A  and  B  to  be  the  two 
points  (Fig.  89).  At  each  of  these  points  is  fixed  a  levelling-staff, 


120 


VESSELS   IN   COMMUNICATION — LEVELS. 


Fig.  89.— Level! 


that  is,  an  upright  rod  divided  into  parts  of  equal  length,  on  which 
slides  a  small  square  board  whose  centre  serves  as  a  mark  for  the 
observer. 

The  level  being  placed  at  an  intermediate  station,  the  observer 
directs  the  line  of  sight  towards  each  levelling-staff,  and  the  mark 
is  raised  or  lowered  till  the  line  of  sight  passes  through  its  centre. 
The  marks  on  the  two  staves  are  in  this  way  brought  to  the  same 
level.  The  staff  in  the  rear  is  then  carried  in  advance  of  the  other} 

the  level  is  again 
placed  between 
the  two,  and  an- 
other observation 
taken.  In  this 
way,  by  noting 
the  division  of 
the  staff  at  which 
the  sliding  mark 
stands  in  each 
case,  the  difference  of  levels  of  two  distant  stations  can  be  deduced 
from  observations  at  a  number  of  intermediate  points. 

For  more  accurate  work,  a  telescope  with  attached  spirit-level 

(§  181)  is  used,  and  the  level- 
ling staff  has  divisions  upon 

x*r*"^^^^^^^^^^^^^^q     it  which  are  read  off  through 

the  telescope. 

180.  Spirit-level.— The 
Fig.  9o.-sPirit-ievei.  spirit-level   is   composed   of 

a  glass  tube  slightly  curved. 

containing  a  liquid,  which  is  generally  alcohol,  and  which  fills  the 
whole  extent  of  the  tube,  except  a  small  space  occupied  by  an  air- 
bubble.  This  tube  is  inclosed  in  a  mounting  which  is  firmly  sup- 
ported on  a  stand. 

Suppose  the  tube  to  have  been  so  constructed  that  a  vertical 

section  of  its  upper  surface  is 
an  arc  of  a  circle,  and  suppose 

-. the  instrument  placed  upon  a 

Fig.  si.  horizontal    plane    (Fig.    91). 

...  The  air-bubble  will  take  up 

a  pos-faon  MN  at  the  highest  part  of  the  tube,  such  that  the 
arcs  MA  and  NB  are  equal.  Hence  it  follows  that  if  the  level 


SPIRIT-LEVEL. 


121 


be  reversed  end  for  end,  the  bubble  will  occupy  the  same  position 

in  the  tube,  the  point  N  coming  to  M,  and  vice  versa.     This  will  not 

be  the  case  if  AB  is  inclined  to  the  horizon  (Fig.  92),  for  then  the 

bubble   will   always   stand 

nearest  to  that  end  of  the 

tube  which  is  highest,  and 

will    therefore    change   its 

place  in  the  tube  when  the  Fig.  92. 

latter  is  reversed.    The  test, 

then,  of  the  horizontality  of  the  line  on  which  the  spirit-level  rests 

is,  that  after  this  operation  of  reversal  the  bubble  should  remain 

between  the  same  marks  on  the  tube.      The  maker  marks  upon 

the   tube   two   points   equidistant    from   the   centre,   the   distance 

between  them  being  equal  to  the  usual  length  of  the  bubble;  and 

the  instrument   ought  to   be  so  adjusted   that  when   the   line  on 

which  it  stands  is  horizontal,  the  ends  of  the  bubble  are  at  these 

marks. 

In  order  that  a  plane  surface  may  be  horizontal,  we  must  have 
two  lines  in  it  horizontal.     This  result  may  be  attained  in  the 


Fig.  93.— Testing  the  Horizontality  of  a  Surface. 

following  manner: — The  body  whose  surface  is  to  be  levelled  is 
made  to  rest  on  three  levelling-screws  which  form  the  three  vertices 
of  an  isosceles  triangle;  the  level  is  first  placed  parallel  to  the  base 
of  the  triangle,  and,  by  means  of  one  of  the  screws,  the  bubble  is 
brought  between  the  reference-marks.  The  instrument  is  then 
placed  perpendicularly  to  its  first  position,  and  the  bubble  is  brought 
between  the  marks  by  means  of  the  third  screw;  this  second  opera- 
tion cannot  disturb  the  result  of  the  first,  since  the  plane  has  only 
been  turned  about  a  horizontal  line  as  hinge. 

181.  Telescope  with  Attached  Level. — In  order  to  apply  the  spirit- 
level  to  land-surveying,  an  apparatus  such  as  that  represented  in 


122 


VESSELS   IN   COMMUNICATION— LEVELS. 


Fig.  94  is  employed.     Upon  a  frame  AA,  movable  about  a  vertical 
axis  B,  are  placed  a  spirit-level  nn,  and  a  telescope  LL,  in  positions 

parallel  to  each  other.  The 
telescope  is  furnished  at  its 
focus  with  two  fine  wires 
crossing  one  another,  whose 
point  of  intersection  deter- 
mines the  line  of  sight  with 
great  precision.  The  appar- 
atus, which  is  provided  with 
levelling-screws  H,  rests  on  a 
tripod  stand,  and  the  observer 
is  able,  by  turning  it  about 
its  axis,  to  command  the  dif- 
ferent points  of  the  horizon.  By  a  process  of  adjustment  which 
need  not  here  be  described,  it  is  known  that  when  the  bubble 
is  between  the  marks  the  line  of  sight  is  horizontal.  By  furnishing 
the  instrument  with  a  graduated  horizontal  circle  P,  we  may  obtain 
the  azimuths  of  the  points  observed,  and  thus  map  out  contour  lines. 
Divisions  are  sometimes  placed  on  each  side  of  the  reference- 
marks  of  the  bubble,  for  measuring  small  deviations  from  horizon- 
tality.  It  is,  in  fact,  easy  to  see,  by  reference  to  Fig.  91,  that  by 
tilting  the  level  through  any  small  angle,  the  bubble  is  displaced  by 
a  quantity  proportional  to  this  angle,  at  least  when  the  curvature 
of  the  instrument  is  that  of  a  circle. 

For  determining  the  angular  value  corresponding  to  each  division 


Fig.  94.— Spirit  level  with  Telescope. 


of  the  tube,  it  is  usual  to  employ  an  apparatus  opening  like  a  pair 
of  compasses  by  a  hinge  C  (Fig.  95),  on  one  of  the  legs  of  which 
rests,  by  two  V-shaped  supports,  the  tube  T  of  the  level.  The  com- 


SPIRIT-LEVEL.  123 

pass  is  opened  by  means  of  a  micrometer  screw  V,  of  very  regular 
action;  and  as  the  distance  of  the  screw  from  the  hinge  is  known, 
as  well  as  the  distance  between  the  threads  of  the  screw,  it  is  easy 
to  calculate  beforehand  the  value  of  the  divisions  on  the  micrometer 
head.  The  levelling-screws  of  the  instrument  serve  to  bring  the 
bubble  between  its  reference-marks,  so  that  the  micrometer  screw  is 
only  used  to  determine  the  value  of  the  divisions  on  the  tube., 


CHAPTER    XVL 


CAPILLARITY. 


182.  Capillarity — General  Phenomena. — The  laws  which  we  have 
thus  far  stated  respecting  the  levels  of  liquid  surfaces  are  subject  to 
remarkable  exceptions  when  the  vessels  in  which  the  liquids  are 
contained  are  very  narrow,  or,  as  they  are  called,  capillary  (capillus, 
a  hair) ;  and  also  in  the  case  of  vessels  of  any  size,  when  we  consider 
the  portion  of  the  liquid  which  is  in  close  proximity  to  the  sides. 

1.  Free  Surface. — The  surface  of  a  liquid  is  not  horizontal  in  the 
neighbourhood  of  the  sides  of  the  vessel,  but  presents  a  very  decided 
curvature.  When  the  liquid  wets  the  vessel,  as  in  the  case  of  water 
in  a  glass  vessel  (Fig.  9G),  the  surface  is  concave;  on  the  contrary 


Fig.  96. 


Fig.  97. 


Fig.  98. 


when  the  liquid  does  not  wet  the  vessel,  as  in  the  case  of  mercury  in 
a  glass  vessel  (Fig.  97),  the  surface  is,  generally  speaking,  convex. 

2.  Capillary  Elevation  and  Depression. — If  a  very  narrow  tube 
of  glass  be  plunged  in  water,  or  any  other  liquid  that  wiU  wet  it 
(Fig.  98),  it  will  be  observed  that  the  level  of  the  liquid,  instead  of 
remaining  at  the  same  height  inside  and  outside  of  the  tube,  stands 
perceptibly  higher  in  the  tube;  a  capillary  ascension  takes  place, 
which  varies  in  amount  according  to  the  nature  of  the  liquid  and 


GENERAL   PHENOMENA.  125 

the  diameter  of  the  tube.  It  will  also  be  seen  that  the  liquid 
column  thus  raised  terminates  in  a  concave  surface.  If  a  glass  tube 
be  dipped  in  mercury,  which  does  not  wet  it,  it  will  be  seen,  by 
bringing  the  tube  to  the  side  of  the  vessel,  that  the  mercury  is 
depressed  in  its  interior,  and  that  it  terminates  in  a  convex  surface 
(Fig.  99). 

3.  Capillary  Vessels  in  Communication  with  Others. — If  we  take 
two  bent  tubes  (Fig.  100),  each  having 
one  branch  of  a  considerable  diameter  and 
the  other  extremely  narrow,  and  pour  into 
one  of  them  a  liquid  which  wets  it,  and 
into  the  other  mercury,  the  liquid  will 
be  observed  in  the  former  case  to  stand 
higher  in  the  capillary  than  in  the  prin- 
cipal branch,  and  in  the  latter  case  to 
stand  lower;  the  free  surfaces  being  at 
the  same  time  concave  in  the  case  of  the 
liquid  which  wets  the  tubes,  and  convex 

Fig.  100.  .        _  ft    i 

in  the  case  of  the  mercury. 

J  183.  Circumstances  which  influence  Capillary  Elevation  and  Depres- 
sion.— In  wetted  tubes  the  elevation  depends  upon  the  nature  of  the 
liquid;  thus,  at  the  temperature  of  18°  Cent.,  water  rises  2979mm  in 
a  tube  1  millimetre  in  diameter,  alcohol  rises  12'18mm,  nitric  acid 
22'57mm,  essence  of  lavender  4'28mm,  &c.  The  nature  of  the  tube  is 
almost  entirely  immaterial,  provided  the  precaution  be  first  taken 
of  wetting  it  with  the  liquid  to  be  employed  in  the  experiment,  so 
as  to  leave  a  film  of  the  liquid  adhering  to  the  sides  of  the  tube. 

Capillary  depression,  on  the  other  hand,  depends  both  on  the 
nature  of  the  liquid  and  on  that  of  the  tube.  Both  ascension  and 
depression  dimmish  as  the  temperature  increases;  for  example,  the 
elevation  of  water,  which  in  a  tube  of  a  certain  diameter  is  equal  to 
132mm  at  0°  Cent.,  is  only  106mm  at  100°. 

_  184.  Law  of  Diameters. — Capillary  elevations  and  depressions, 
when  all  other  circumstances  are  the  same,  are  inversely  propor- 
tional to  the  diameters  of  the  tubes.  As  this  law  is  a  consequence 
of  the  mathematical  theories  which  are  generally  accepted  as  ex- 
plaining capillary  phenomena,  its  verification  has  been  regarded  as 
of  great  importance. 

The  experiments  of  Gay-Lussac,  which  confirmed  this  law,  have 
been  repeated,  with  slight  modifications,  by  several  observers.  The 


126 


CAPILLARITY. 


method  employed  consists  essentially  in  measuring  the  capillary 
elevation  of  a  liquid  by  means  of  a  cathetometer  (Fig.  101).  The 
telescope  II  is  directed  first  to  the  top  n  of  the  column  in  the  tube, 
and  then  to  the  end  of  a  pointer  b,  which  touches  the  surface  of  the 


Fig.  101.— Verification  of  Law  of  Diameters. 

liquid  at  a  point  where  it  is  horizontal.  In  observing  the  depression 
of  mercury,  since  the  opacity  of  the  metal  prevents  us  from  seeing 
the  tube,  we  must  bring  the  tube  close  to  the  side  of  the  vessel  e. 

The  diameter  of  the  tube  can  be  measured  directly  by  observing 
its  section  through  a  microscope,  or  we  may  proceed  by  the  method 
employed  by  Gay-Lussac.  He  weighed  the  quantity  of  mercury 
which  filled  a  known  length  I  of  the  tube;  this  weight  w  is  that 
of  a  cylinder  of  mercury  whose  radius  x  is  determined  by  the 
equation  13'59  irxH—w,  where  x  and  I  are  in  centimetres,  and  w  in 
grammes. 

The  result  of  these  different  experiments  is,  that  in  the  case  of 
wetted  tubes  the  law  is  exactly  fulfilled,  provided  that  they  be  pre- 
viously washed  with  the  greatest  care,  so  as  to  remove  all  foreign 
matters,  and  that  the  liquid  on  which  the  experiment  is  to  be  per- 
formed be  first  passed  through  them.  When  the  liquid  does  not  wet 
the  tube,  various  causes  combine  to  affect  the  form  of  the  surface  in 
which  the  liquid  column  terminates;  and  we  cannot  infer  the  depres- 
sion from  knowing  the  diameter,  unless  we  also  take  into  considera- 
tion some  element  connected  with  the  form  of  the  terminal  surface, 
such  as  the  length  of  the  sagitta,  or  the  angle  made  with  the  sides 


FUNDAMENTAL   PRINCIPLES.  127 

of  the  tube  by  the  extremities  of  the  curved  surface,  which  is  called 
the  angle  of  contact. 

-  185.  Fundamental  Laws  of  Capillary  Phenomena. — Capillary  phe- 
nomena, as  they  take  place  alike  in  air  and  in  vacuo,  cannot  be  attri- 
buted to  the  action  of  the  atmosphere.  They  depend  upon  molecular 
actions  which  take  place  between  the  particles  of  the  liquid  itself, 
and  between  the  liquid  and  the  solid  containing  it,  the  actions  in 
question  being  purely  superficial — that  is  to  say,  being  confined  to 
an  extremely  thin  layer  forming  the  external  boundary  of  the  liquid, 
and  to  an  extremely  thin  superficial  layer  of  the  solid  in  contact 
with  the  liquid.  For  example,  it  is  found  in  the  case  of  glass  tubes, 
that  the  amount  of  capillary  elevation  or  depression  is  not  at  all 
affected  by  the  thickness  of  the  sides  of  the  tube.  The  following 
are  some  of  the  principles  which  govern  capillary  phenomena. 

1.  For  a  given  liquid  in  contact  with  a  given  solid,  with  a  definite 
intimateness  of  contact'  (this  last  element  being  dependent  upon  the 
cleanness  of  the  surface,  upon  whether  the  surface  of  the  solid  has 
been  recently  washed  by  the  liquid,  and  perhaps  upon  some  other 
particulars),  there  is  (at  any  specified  temperature)  a  definite  angle 
of  contact,  which  is  independent  of  the  directions  of  the  surfaces 
with  regard  to  the  vertical. 

2.  Every  liquid  behaves  as  if  a  thin  film,  forming  its  external 
layer,  were  in  a  state  of  tension,  and  exerting  a  constant  effort  to 
contract.     This  tension,  or  contractile  force,  is 'exhibited  over  the 
whole  of  the  free  surface  (that  is,  the  surface  which  is  exposed  to  air); 
but  wherever  the  liquid  is  in  contact  with  a  solid,  its  existence  is 
masked  by  other  molecular  actions.     It  is  uniform  in  all  directions 
in  the  free  surface,  and  at  all  points  in  this  surface,  being  dependent 
only  on  the  nature  and  temperature  of  the  liquid.     Its  intensity  for 
several  specified  liquids  is  given  in  tabular  form  further  on  (§  192) 
upon  the  authority  of  Van  der  Mensbrugghe.     Tension  of  this  kind 
must  of  course  be  stated  in  units  of  force  per  linear  unit,  because  by 
doubling  the  width  of  a  band  we  double  the  force  required  to  keep 
it  stretched.     Mensbrugghe  considers  that  such  tension  really  exists 
in  the  superficial  layer;  but  the  majority  of  authors  (and  we  think 
with  more  justice)  regard  it  rather  as  a  convenient  fiction,  which 
accurately  represents  the  effects  of  the  real  cause.     Two  of  the  most 
eminent  writers  on  the  cause  of  capillary  phenomena  are  Laplace 
and  Dr.  Thomas  Young.     The  subject  presents  difficulties   which 
have  not  yet  been  fully  surmounted. 


128  CAPILLARITY. 

186.  Application  to  Elevation  in  Tubes. — The  law  of  diameters  is 
a  direct  consequence  of  the  two  preceding  principles;  for  if  a  denote 
the  external  angle  of  contact  (which  is  acute  in  the  case  of  mercury 
against  glass),  T  the  tension  per  unit  length,  and  r  the  radius  of 
the  tube,  then  2irrT  will  be  the  whole  amount  of  force  exerted  at 
the  margin  of  the  surface;  and  as  this  force  is  exerted  in  a  direction 
making  an  angle  a  with  the  vertical,  its  vertical  component  will  be 
2irrT  cos  a,  which  is  exerted  in  pulling  the  tube  upwards  and  the 
liquid  downwards. 

If  w  be  the  weight  of  unit  volume  of  the  liquid,  then  irr~w  is  the 
weight  of  as  much  as  would  occupy  unit  length  of  the  tube;  and  if  h 
denote  the  height  of  a  column  whose  weight  is  equal  to  the  force 
tending  to  depress  the  liquid,  we  have 

ir>shw  -  2irrT  cos  a ; 

whence  h=  rcf^a»  which,  when  the  other  elements  are  given,  varies 
inversely  as  r,  the  radius  of  the  tube. 

Having  regard  to  the  fact  that  the  surface  is  not  of  the  same 
height  in  the  centre  as  at  the  edges,  it  is  obvious  that  h  denotes  the 
mean  height. 

If  a  be  obtuse,  h  will  be  negative— that  is  to  say,  there  will  be 
elevation  instead  of  depression.  In  the  case  of  water  against  a  tube 
which  has  been  well  wetted  with  that  liquid,  a  is  180°— that  is  to 
say,  the  tube  is  tangential  to  the  surface.  For  this  case  the  formula 
for  h  gives 


Again,  for  two  parallel  vertical  plates  at  distance  u,  the  vertical  force 
of  capillarity  for  a  unit  of  length  is  2Tco£a,  which  must  be  equal  to 
whu,  being  the  weight  of  a  sheet  of  liquid  of  height  h,  thickness  u, 
and  length  unity.  We  have  therefore 

ft  _  2Tcosa 
uw    ' 

which  agrees  with  the  expression  for  the  depression  or  elevation  in 
a  circu  ar  tube  whose  radius  is  equal  to  the  distance  between  these 
parallel  plates. 

The  surface  tension  always  tends  to  reduce  the  surface  to  the 
smallest  area  which  can  be  inclosed  by  its  actual  boundary;  and 
herefore  always  produces  a  normal  force  directed  from  the  convex 
to  the  concave  side  of  the  superficial  film.  Hence,  wherever  there  is 


PRESSURE   EXERTED   BY   FILM.  129 

capillary  elevation  the  free  surface  must  be  concave;  wherever  there 
is  depression  it  must  be  convex. 

187.  It  follows  from  a  well-known  proposition  in  statics  (Tod- 
hunter's  Statics,  §  194),  that  if  a  cylindrical  film  be  stretched  with  a 
uniform  tension  T  (so  that  the  force  tending  to  pull  the  film  asunder 
across  any  short  line  drawn  on  the  film,  is  T  times  the  length  of  the 
line),  the  resultant  normal  pressure  (which  the  film  exerts,  for  ex- 
ample, against  the  surface  of  a  solid  internal  cylinder  over  which  it 
is  stretched)  is  T  divided  by  the  radius  of  the  cylinder. 

It  can  be  proved  that  a  film  of  any  form,  stretched  with  uniform 
tension  T,  exerts  at  each  point  a  normal  pressure  equal  to  the  sum 
of  the  pressures  which  would  be  exerted  by  two  overlapping  cylin- 
drical films,  whose  axes  are  at  right  angles  to  one  another,  and 
whose  cross  sections  are  circles  of  curvature  of  normal  sections  at 
the  point.  That  is  to  say,  if  P  be  the  normal  force  per  unit  area, 
and  r,  r'  the  radii  of  curvature  in  two  mutually  perpendicular  normal 
sections  at  the  point,  then 

'-»£+*> 

At  any  point  on  a  curved  surface,  the  normal  sections  of  greatest  and 
least  curvature  are  mutually  perpendicular,  and  are  called  the  prin- 
cipal normal  sections  at  the  point.  If  the  corresponding  radii  of 
curvature  be  R,  R',  we  have 


or  the  normal  force  per  unit  area  is  equal  to  the  tension  per  unit 
length  multiplied  by  the  sum  of  the  principal  curvatures, 

In  the  case  of  capillary  depressions  and  elevations,  the  superficial 
film  at  the  free  surface  is  to  be  regarded  as  pressing  the  liquid  in- 
wards, or  pulling  it  outwards,  according  as  this  surface  is  convex  or 
concave,  with  a  force  P  given  by  the  above  formula.  The  value  of 
P  at  any  point  of  the  free  surface  is  equal  to  the  pressure  due  to  the 
height  of  a  column  of  liquid  extending  from  that  point  to  the  level 
of  the  general  horizontal  surface.  It  is  therefore  greatest  at  the 
edges  of  the  elevated  or  depressed  column  in  a  tube,  and  least  in  the 

centre;  and  the  curvature,  as  measured  by  ^  +  ~n  must  vary  in  the 

same  proportion.  If  the  tube  is  so  large  that  there  is  no  sensible 
elevation  or  depression  in  the  centre  of  the  column,  the  centre  of  the 
free  surface  must  be  sensibly  plane. 

188.  Another  consequence  of  the  formula  is,  that  in  circumstances 


130  CAPILLARITY. 

where  there  can  be  no  normal  pressure  towards  either  side  of  the 
surface, 

which  implies  that  either  the  surface  is  plane,  in  which  case  each  of 
the  two  terms  is  separately  equal  to  zero,  or  else 

E  =  -  K';  (3) 

that  is,  the  principal  radii  of  curvature  are  equal,  and  lie  on  opposite 
sides  of  the  surface.  The  formulae  (2),  (3)  apply  to  a  film  of  soapy 
water  attached  to  a  loop  of  wire.  If  the  loop  be  in  one  plane,  the 
film  will  be  in  the  same  plane.  If  the  loop  be  not  in  one  plane,  the 
film  cannot  be  in  one  plane,  and  will  in  fact  assume  that  form  which 
gives  the  least  area  consistent  with  having  the  loop  for  its  boundary. 
At  every  point  it  will  be  observed  to  be,  if  we  may  so  say,  concave 
towards  both  sides,  and  convex  towards  both  sides,  the  concavity 
being  precisely  equal  to  the  convexity — that  is  to  say,  equation  (3) 
is  satisfied  at  every  point  of  the  film. 

In  this  case  both  sides  of  the  film  are  exposed  to  atmospheric 
pressure.  In  the  case  of  a  common  soap-bubble  the  outside  is  ex- 
posed to  atmospheric  pressure,  and  the  inside  to  a  pressure  somewhat 
greater,  the  difference  of  the  pressures  being  balanced  by  the  ten- 
dency of  the  film  to  contract.  Formula  (1)  becomes  for  either  the 
outer  or  inner  surface  of  a  spherical  bubble 


but  this  result  must  be  doubled,  because  there  are  two  free  surfaces; 
hence  the  excess  of  pressure  of  the  inclosed  above  the  external  air  is 

jrp 

-£,  R  denoting  the  radius  of  the  bubble. 

The  value  of  T  for  soapy  water  is  about  1  grain  per  linear  inch; 
hence,  if  we  divide  4  by  the  radius  of  the  bubble  expressed  in  inches, 
we  shall  obtain  the  excess  of  internal  over  external  pressure  in  grains 
per  square  inch. 

The  value  of  T  for  any  liquid  may  be  obtained  by  observing  the 
amount  of  elevation  or  depression  in  a  tube  of  given  diameter,  and 
employing  the  formula 


which  follows  immediately  from  the  formula  for  h  in  §  ISO. 

189.  It  is  this  uniform  surface  tension,  of  which  we  have  been 


DROPS.  131 

speaking,  which  causes  a  drop  of  a  liquid  falling  through  the  air 
either  to  assume  the  spherical  form,  or  to  oscillate  about  the  spheri- 
cal form.  The  phenomena  of  drops  can  be  imitated  on  an  enlarged 
scale,  under  circumstances  which  permit  us  to  observe  the  actual 
motions,  by  a  method  devised  by  Professor  Plateau  of  Ghent.  Olive- 
oil  is  intermediate  in  density  between  water  and  alcohol.  Let  a 
mixture  of  alcohol  and  water  be  prepared,  having  precisely  the 
density  of  olive-oil,  and  let  about  a  cubic  inch  of  the  latter  be  gently 
introduced  into  it  with  the  aid  of  a  funnel  or  pipette.  It  will  as- 
sume a  spherical  form,  and  if  forced  out  of  this  form  and  then  left 
free,  will  slowly  oscillate  about  it;  for  example,  if  it  has  been  com- 
pelled to  assume  the  form  of  a  prolate  spheroid,  it  will  pass  to  the 
oblate  form,  will  then  become  prolate  again,  and  so  on  alternately, 
becoming  however  more  nearly  spherical  every  time,  because  its 
movements  are  hindered  by  friction,  until  at  last  it  comes  to  rest  as 
a  sphere. 

190.  Capillarity  furnishes  no  exception  to  the  principle  that  the 
pressure  in  a  liquid  is  the  same  at  all  points  at  the  same  depth. 
When  the  free  surface  within  a  tube  is  convex,  and  is  consequently 
depressed  below  the  plane  surface  of  the  external  liquid,  the  pres- 
sure becomes  suddenly  greater  on  passing  downwards  through  the 
superficial  layer,  by  the  amount  due  to  the  curvature.  Below  this 
it  increases  regularly  by  the  amount  due  to  the  depth  of  liquid 
passed  through.  The  pressure  at  any  point  vertically  under  the  con- 
vex meniscus1  may  be  computed,  either  by  taking  the  depth  of  the 
point  below  the  general  free  surface,  and  adding  atmospheric  pres- 
sure to  the  pressure  due  to  this  depth,  according  to  the  ordinary- 
principles  of  hydrostatics,  or  by  taking  the  depth  of  the  point  below 
that  point  of  the  meniscus  which  is  vertically  over  it,  adding  the 
pressure  due  to  the  curvature  at  this  point,  and  also  adding  atmo- 
spheric pressure. 

When  the  free  surface  of  the  liquid  within  a  tube  is  concave,  the 
pressure  suddenly  diminishes  on  passing  downwards  through  the 
superficial  layer,  by  the  amount  due  to  the  curvature  as  given  by 
formula  (1);  that  is  to  say,  the  pressure  at  a  very  small  depth  is  less 
than  atmospheric  pressure  by  this  amount.  Below  this  depth  it 
goes  on  increasing  according  to  the  usual  law,  and  becomes  equal  to 

1  The  convex  or  concave  surface  of  the  liquid  in  a  tube  is  usually  denoted  by  the  name 
meniscus  (ftyviffKos,  a  crescent),  which  denotes  a  form  approximately  resembling  that  of  a 
watch-glass. 


132  CAPILLAEITY. 

atmospheric  pressure  at  that  depth  which  corresponds  with  the  level 
of  the  plane  external  surface.  The  pressure  at  any  point  in  the 
liquid  within  the  tube  can  therefore  be  obtained  either  by  subtract- 
ing from  atmospheric  pressure  the  pressure  due  to  the  elevation  of 
the  point  above  the  general  surface,  or  by  adding  to  atmospheric 
pressure  the  pressure  due  to  the  depth  below  that  point  of  the 
meniscus  which  is  on  the  same  vertical,  and  subtracting  the  pressure 
due  to  the  curvature  at  this  point. 

These  rules  imply,  as  has  been  already  remarked,  that  the  curva- 
ture is  different  at  different  points  of  the  meniscus,  being  greatest 
where  the  elevation  or  depression  is  greatest,  namely  at  the  edges 
of  the  meniscus;  and  least  at  the  point  of  least  elevation  or  depres- 
sion, which  in  a  cylindrical  tube  is  the  middle  point. 

The  principles  just  stated  apply  to  all  cases  of  capillary  elevation 
and  depression. 

They  enable  us  to  calculate  the  force  with  which  two  parallel  ver- 
tical plates,  partially  immersed  in  a  liquid  which  wets  them,  are 
urged  towards  each  other  by  capillary  action.  The  pressure  in  the 
portion  of  liquid  elevated  between  them  is  less  than  atmospheric, 
and  therefore  is  insufficient  to  balance  the  atmospheric  pressure 
which  is  exerted  on  the  outer  faces  of  the  plates.  The  average  pres- 
sure in  the  elevated  portion  of  liquid  is  equal  to  the  actual  pressure 
at  the  centre  of  gravity  of  the  elevated  area,  and  is  less  than  atmo- 
spheric pressure  by  the  pressure  of  a  column  of  liquid  whose  height 
is  the  elevation  of  this  centre  of  gravity. 

Even  if  the  liquid  be  one  which  does  not  wet  the  plates,  they  will 
still  be  urged  towards  each  other  by  capillary  action;  for  the  inner 
faces  of  the  plates  are  exposed  to  merely  atmospheric  pressure  over 
the  area  of  depression,  while  the  corresponding  portions  of  the  ex- 
ternal faces  are  exposed  to  atmospheric  pressure  increased  by  the 
weight  of  a  portion  of  the  liquid. 

These  principles  explain  the  apparent  attraction  exhibited  by 
bodies  floating  on  a  liquid  which  either  wets  them  both  or  wets 
neither  of  them.  When  the  two  bodies  are  near  each  other  they 
behave  somewhat  like  parallel  plates,  the  elevation  or  depression  of 
the  liquid  between  them  being  greater  than  on  their  remote  sides. 

If  two  floating  bodies,  one  of  which  is  wetted  and  the  other  un- 
wetted  by  the  liquid,  come  near  together,  the  elevation  and  depres- 
sion of  the  liquid  will  be  less  on  the  near  than  on  the  remote  sides, 
and  apparent  repulsion  will  be  exhibited. 


APPARENT   ATTRACTIONS.  133 

In  all  cases  of  capillary  elevation  or  depression,  the  solid  is  pulled 
downwards  or  upwards  with  a  force  equal  to  that  by  which  the 
liquid  is  raised  or  depressed.  In  applying  the  principle  of  Archi- 
medes to  a  solid  partially  immersed  in  a  liquid,  it  is  therefore  neces- 
sary (as  we  have  seen  in  §  159),  when  the  solid  produces  capillary 
depression,  to  reckon  the  void  space  thus  created  as  part  of  the  dis- 
placement; and  when  the  solid  produces  capillary  elevation,  the  fluid 
raised  above  the  general  level  must  be  reckoned  as  negative  displace- 
ment, tending  to  increase  the  apparent  weight  of  the  solid. 

191.  Thus  far  all  the  effects  of  capillary  action  which  we  have 
mentioned  are  connected  with  the  curvature  of  the  superficial  film, 
and  depend  upon  the  principle  that  a  convex  surface  increases  and  a 
concave  surface  diminishes  the  pressure  in  the  interior  of  the  liquid. 
But  there  is  good  reason  for  maintaining  that  whatever  be  the  form 
of  the  free  surface  there  is  always  pressure  in  the  interior  due  to 
the  molecular  action  at  this  surface,  and  that  the  pressure  due  to  the 
curvature  of  the  surface  is  to  be  added  to  or  subtracted   from  a 
definite  amount  of  pressure  which  is  independent  of  the  curvature 
and  depends  only  on  the  nature  and  condition  of  the  liquid.     This 
indeed  follows  at  once  from  the  fact  that  capillary  elevation  can 
take  place  in  vacuo.     As  far  as  the  principles  of  the  preceding 
paragraphs  are  concerned,  we  should  have,  at  points  within  the 
elevated  column,  a  pressure  less  than  that  existing  in  the  vacuum. 
This,  however,  cannot  be;  we  cannot  conceive  of  negative  pressure 
existing  in  the  interior  of  a  liquid,  and  we  are  driven  to  conclude 
that  the  elevation  is  owing  to  the  excess  of  the  pressure  caused  by 
the  plane  surface  in  the  containing  vessel  above  the  pressure  caused 
by  the  concave  surface  in  the  capillary  tube. 

There  are  some  other  facts  which  seem  only  explicable  on  the  same 
general  principle  of  interior  pressure  due  to  surface  action, — facts 
which  attracted  the  notice  of  some  of  the  earliest  writers  on 
pneumatics,  namely,  that  siphons  will  work  in  vacuo,  and  that  a 
column  of  mercury  at  least  75  inches  in  length  can  be  sustained — as 
if  by  atmospheric  pressure — in  a  barometer  tube,  the  mercury  being 
boiled  and  completely  filling  the  tube. 

192.  We  have  now  to  notice  certain  phenomena  which  depend  on 
the  difference  in  the  surface  tensions  of  different  liquids,  or  of  the 
same  liquid  in  different  states. 

Let  a  thin  layer  of  oil  be  spread  over  the  upper  surface  of  a  thin 
sheet  of  brass,  and  let  a  lamp  be  placed  underneath.  The  oil  will  be 


134 


CAPILLARITY. 


observed  to  run  away  from  the  spot  directly  over  the  flame,  even 
though  this  spot  be  somewhat  lower  than  the  rest  of  the  sheet. 
This  Sf&et  is  attributable  to  the  excess  of  surface  tension  in  the  cold 
oil  above  the  hot. 

In  like  manner,  if  a  drop  of  alcohol  be  introduced  into  a  thin 
layer  of  water  spread  over  a  nearly  horizontal  surface,  it  will  be 
drawn  away  in  all  directions  by  the  surrounding  water,  leaving  a 
nearly  dry  spot  in  the  space  which  it  occupied.  In  this  experiment 
the  water  should  be  coloured  in  order  to  distinguish  it  from  the 
alcohol. 

Again,  let  a  very  small  fragment  of  camphor  be  placed  on  the  sur- 
face of  hot  water.  It  will  be  observed  to  rush  to  and  fro,  with 
frequent  rotations  on  its  own  axis,  sometimes  in  one  direction  and 
sometimes  in  the  opposite.  These  effects,  which  have  been  a  frequent 
subject  of  discussion,  are  now  known  to  be  due  to  the  diminution  of 
the  surface  tension  of  the  water  by  the  camphor  which  it  takes  up. 
Superficial  currents  are  thus  created,  radiating  from  the  fragment  of 
camphor  in  all  directions;  and  as  the  camphor  dissolves  more  quickly 
in  some  parts  than  in  others,  the  currents  which  are  formed  are  not 
equal  in  all  directions,  and  those  which  are  most  powerful  prevail 
over  the  others  and  give  motion  to  the  fragment. 

The  values  of  T,  the  apparent  surface  tension,  for  several  liquids, 
are  given  in  the  following  table,  on  the  authority  of  Van  der  Mens- 
brugghe,  in  milligrammes  (or  thousandth  parts  of  a  gramme)  per 
millimetre  of  length.  They  can  be  reduced  to  grains  per  inch  of 
length  by  multiplying  them  by  "392;  for  example,  the  surface  ten- 
sion of  distilled  water  is  7*3  X  '392  =  2-86  grains  per  inch. 


Distilled  water  at  20°  Cent 7'3 

Sulphuric  ether, ]  -88 

Absolute  alcohol, ,     2'5 

Olive-oil, 3-5 

Mercury, 49-1 

Bisulphide  of  carbon 3 -5  7 


Solution  of  Marseilles  soap,  1  part  of 

soap  to  40  of  water 2 '83 

Solution  of  saponine, 4 '67 

Saturated  solution  of  carbonate  of 

soda, 4-28 

Water  impregnated  with  camphor,  .  4 '5 


193.  Endosmose. — Capillary  phenomena  have  undoubtedly  some 
connection  with  a  very  important  property  discovered  by  Dutrochet, 
and  called  by  him  endosmose. 

The  endosmometer  invented  by  him  to  illustrate  this  phenomenon 
consists  of  a  reservoir  v  (Fig.  102)  closed  below  by  a  membrane  la, 
and  terminating  above  in  a  tube  of  considerable  length.  This  reser- 
voir is  filled,  suppose,  with  a  solution  of  gum  in  water,  and  is  kept 


DIFFUSION  THROUGH   SEPTA. 


135 


immersed  in  water.  At  the  end  of  some  time  the  level  of  the  liquid 
in  the  tube  will  be  observed  to  have  risen  to  n,  suppose,  and  at  the 
same  time  traces  of  gum  will  be  found  in  the  water  in  which  the 
reservoir  is  immersed.  Hence  we  conclude  that  the  two  liquids 
have  penetrated  through  the  membrane,  but  in  different  proportions ; 
and  this  is  what  is  called  endosmose. 

If  instead  of  a  solution  of  gum  we  employed  water  containing 
albumen,  sugar,  or  gelatine  in  solution,  a  similar  result  would  ensue. 
The  membrane  may  be  replace/!  by  a  slab  of  wood  or  of  porous  clay. 
Physiologists  have  justly  attached  very  great  importance  to  this 
discovery  of  Dutrochet.  It  explains,  in  fact,  the  interchange  of 
liquids  which  is  continually  taking  place  in  the  tissues  and  vessels 
of  the  animal  system,  as  well  as  the  absorption  of  water  by  the 
spongioles  of  roots,  and  several  similar  phenomena. 

As  regards  the  power  of  passing  through  porous  diaphragms, 
Graham  has  divided  substances  into  two  classes — crystalloids  and 
colloids  (K('i\\r),  glue).  The  former  are  sus- 
ceptible of  crystallization,  form  solutions  free 
from  viscosity,  are  sapid,  and  possess  great 
powers  of  diffusion  through  porous  septa. 
The  latter,  including  gum,  starch,  albumen, 
&c.,  are  characterized  by  a  remarkable  slug- 
gishness and  indisposition  both  to  diffusion 
and  to  crystallization,  and  when  pure  are 
nearly  tasteless. 

Diffusion  also  takes  place  through  col- 
loidal diaphragms  which  are  not  porous, 
the  diaphragm  in  this  case  acting  as  a 
solvent,  and  giving  out  the  dissolved  mate- 
rial on  the  other  side.  In  the  important 
process  of  modern  chemistry  called  dialysis, 
saline  ingredients  are  separated  from  or- 
ganic substances  with  which  they  are 
blended,  by  interposing  a  colloidal  dia- 
phragm (De  La  Rue's  parchment  paper) 
between  the  mixture  and  pure  water. 
The  organic  matters,  being  colloidal,  remain 

behind,  while  the  salts  pass  through,  and  can  be  obtained  in  a 
nearly  pure  state  by  evaporating  the  water. 

Gases  are  also  capable  of  diffusion  through  diaphragms,  whether 


Fig.  102.— Endosmometer. 


13C  CAPILLARITY. 

porous  or  colloidal,  the  rate  of  diffusion  being  in  the  former  case 
inversely  as  the  square  root  of  the  density  of  the  gas.  Hydrogen 
diffuses  so  rapidly  through  unglazed  earthenware  as  to  afford  oppor- 
tunity for  very  striking  experiments;  and  it  shows  its  power  of 
traversing  colloids  by  rapidly  escaping  through  the  sides  of  india- 
rubber  tubes,  or  through  films  of  soapy  water. 


CHAPTER    XVII. 


THE   BAROMETER. 


.  194.  Expansibility  of  Gases. — Gaseous  bodies  possess  a  number  of 
properties  in  common  with  liquids;  like  them,  they  transmit  pres- 
sures entire  and  in  all  directions,  according  to  the  principle  of 
Pascal;  but  they  differ  essentially  from  liquids  in  the  permanent 
repulsive  force  exerted  between  their  molecules,  in  virtue  of  which 
a  mass  of  gas  always  tends  to  expand. 

This  property,  called  the  expansibility  of  gases,  is  commonly  illus- 
trated by  the  following  experiment: — 

A  bladder,  nearly  empty  of  air,  and  tied  at  the  neck,  is  placed 
under  the  receiver  of  an 
air-pump.  At  first  the 
air  which  it  contains 
and  the  external  air 
oppose  each  other  by 
their  mutual  pressure, 
and  are  in  equilibrium. 
But  if  we  proceed  to 
exhaust  the  receiver, 
and  thus  diminish  the 
external  pressure,  the 
bladder  gradually  be- 
comes inflated,  and  thus 
manifests  the  tendency 
of  the  gas  which  it  con- 
tains tO  OCCUpy  a  greater  Fi«-  lOS.-ExpansibiUty  of  Gases. 

volume. 

However  large  a  vessel  may  be,  it  can  always  be  filled  by  any 
quantity  whatever  of  a  gas,  which  will  always  exert  pressure  against 


138  THE   BAROMETER. 

the  sides.  In  consequence  of  this  property,  the  name  of  elastic 
fluids  is  often  given  to  gases. 

195.  Air  has  Weight. — The  opinion  was  long  held  that  the  air  was 
without  weight;  or,  to  speak  more  precisely,  it  never  occurred  to 
any  of  the  philosophers  who  preceded  Galileo  to  attribute  any 
influence  in  natural  phenomena  to  the  weight  of  the  air.  And  as 
this  influence  is  really  of  the  first  importance,  and  comes  into  play 
in  many  of  the  commonest  phenomena,  it  very  naturally  happened 
that  the  discovery  of  the  weight  of  air  formed  the  commencement 
of  the  modern  revival  of  physical  science. 

It  appears,  however,  that  Aristotle  conceived  the  idea  of  the 
possibility  of  air  having  weight,  and,  in  order  to  convince  himself 
on  this  point,  he  weighed  a  skin  inflated  and  collapsed.  As  he 
obtained  the  same  weight  in  both  cases,  he  relinquished  the  idea 
which  he  had  for  the  moment  entertained.  In  fact,  the  experiment, 
as  he  performed  it,  could  only  give  a  negative  result;  for  if  the 
weight  of  the  skin  was  increased,  on  the  one  hand,  by  the  intro- 
duction of  a  fresh  quantity  of  air,  it  was  diminished,  on  the  other, 
by  the  corresponding  increase  in  the  upward  pressure  of  the  air 
displaced.  In  order  to  draw  a  certain  conclusion,  the  experiment 
should  be  performed  with  a  vessel  which  could  receive  within  it 
air  of  different  degrees  of  density,  without  changing  its  own 
volume. 

Galileo  is  said  to  have  devised  the  experiment  of  weighing  a 
globe  filled  alternately  with  ordinary  air  and  with  compressed  air. 
As  the  weight  is  greater  in  the  latter  case,  Galileo  should  have 
drawn  the  inference  that  air  is  heavy.  It  does  not  appear,  however, 
that  the  importance  of  this  conclusion  made  much  impression  on 
him,  for  he  did  not  give  it  any  of  those  developments  which  might 
have  been  expected  to  present  themselves  to  a  mind  like  his. 

Otto  Guericke,  the  illustrious  inventor  of  the  air-pump,  in  1G50 
performed  the  following  experiment,  which  is  decisive: — 

A  globe  of  glass  (Fig.  104),  furnished  with  a  stop-cock,  and  of 
a  sufficient  capacity  (about  twelve  litres),  is  exhausted  of  air.  It  is 
then  suspended  from  one  of  the  scales  of  a  balance,  and  a  weight 
sufficient  to  produce  equilibrium  is  placed  in  the  other  scale.  The 
stop-cock  is  then  opened,  the  air  rushes  into  the  globe,  and  the  beam 
is  observed  gradually  to  incline,  so  that  an  additional  weight  is 
required  in  the  other  scale,  in  order  to  re-establish  equilibrium.  If 
the  capacity  of  the  globe  is  12  litres,  about  15'5  grammes  will  be 


WEIGHT   OF  AIR. 


139 


needed,  which  gives  1*3  gramme  as  the  approximate   weight   of 
a  litre  (or  1000  cubic  centimetres) 
of  air.1 

If,  in  performing  this  experiment, 
we  take  particular  precautions  to 
insure  its  precision,  as  we  shall 
explain  in  the  book  on  Heat,  it  will 
be  found  that,  at  the  temperature 
of  freezing  water,  and  under  the 
pressure  of  one  atmosphere,  a  litre 
of  perfectly  dry  air  wreighs  1*293 
gramme.2  Under  these  circum- 
stances, the  ratio  of  the  weight  of 
a  volume  of  air  to  that  of  an  equal 

volume  of  water  is  l|S=-JL     Air 

1UUU        /  /o 

is  thus  773  times  lighter  than  water. 
By  repeating  this  experiment 
with  other  gases,  we  may  determine 
their  weight  as  compared  with  that 
of  air,  and  the  absolute  weight  of  a 
litre  of  each  of  them.  Thus  it  is 
found  that  a  litre  of  oxygen  weighs 
1-43  gramme,  a  litre  of  carbonic 
hydrogen  0'089  gramme,  &c. 


Fig.  104.— Weight  of  Air. 


acid   T97   gramme,  a   litre   of 


1  A  cubic  foot  of  air  in  ordinary  circumstances  weighs  about  an  ounce  and  a  quarter. 

2  In  strictness,  the  weight  in  grammes  of  a  litre  of  air  under  the  pressure  of  760 
millimetres  of  mercury  is  different  in  different  localities,  being  proportional  to  the  inten- 
sity of  gravity — not  because  the  force  of  gravity  on  the  litre  of  air  is  different,  for 
though  this  is  true,  it  does  not  affect  the  numerical  value  of  the  weight  when  stated  in 
grammes,  but  because  the  pressure  of  760  millimetres  of  mercury  varies  as  the  intensity 
of  gravity,  so  that  more  air  is  compressed  into  the  space  of  a  litre  as  gravity  increases. 
(§  201,  6.) 

The  weight  in  grammes  is  another  name  for  the  mass.  The  force  of  gravity  on  a  litre 
of  air  under  the  pressure  of  760  millimetres  is  proportional  to  the  square  of  the  intensity 
of  gravity. 

This  is  an  excellent  example  of  the  ambiguity  of  the  word  weight,  which  sometimes 
denotes  a  mass,  sometimes  a  force ;  and  though  the  distinction  is  of  no  practical  importance 
so  long  as  we  confine  our  attention  to  one  locality,  it  cannot  be  neglected  when  different 
localities  are  compared. 

Eegnault's  determination  of  the  weight  of  a  litre  of  dry  air  at  0°  Cent,  under  the 
pressure  of  760  millimetres  at  Paris  is  1-293187  gramme.  Gravity  at  Paris  is  to  gravity 
at  Greenwich  as  3456  to  3457.  The  corresponding  number  for  Greenwich  is  therefore 
1-293561. 


140 


THE   BAROMETER. 


196.  Atmospheric  Pressure. — The  atmosphere  encircles  the  earth 
with  a  layer  some  50  or  100  miles  in  thickness;  this  heavy  fluid 
mass  exerts  on  the  surface  of  all  bodies  a  pressure  entirely  analogous 
both  in  nature  and  origin  to  that  sustained  by  a  body  wholly 
immersed  in  a  liquid.  It  is  subject  to  the  fundamental  laws  men- 
tioned in  §§  137-139.  The  pressure  should  therefore  diminish  as 
we  ascend  from  the  surface  of  the  earth,  but  should  have  the  same 
value  for  all  points  in  the  same  horizontal  layer,  provided  that  the 
air  is  in  a  state  of  equilibrium.  On  account  of  the  great  compressi- 
bility of  gas,  the  lower  layers  are  much  more  dense  than  the  upper 
ones;  but  the  density,  like  the  pressure,  is  constant  in  value  for  the 


Fig.  105.— Torricellian  Exj 


same  horizontal  layer,  throughout  any  portion  of  air  in  a  state  of 
equilibrium.  Whenever  there  is  an  inequality  either  of  density  or 
pressure  at  a  given  level,  wind  must  ensue. 


PRESSURE   OF   ONE   ATMOSPHERE. 


141 


We  owe  to  Torricelli  an  experiment  which  plainly  shows  the 
pressure  of  the  atmosphere,  and  enables  us  to  estimate  its  intensity 
with  great  precision.  This  experiment,  which  was  performed  in 
1643,  one  year  after  the  death  of  Galileo,  at  a  time  when  the  weight 
and  pressure  of  the  air  were  scarcely  even  suspected,  has  immor- 
talized the  name  of  its  author,  and  has  exercised  a  most  important 
influence  upon  the  progress  of  natural  philosophy. 

197.  Torricellian  Experiment. — A  glass  tube  (Fig.  105)  about  a 
quarter  or  a  third  of  an  inch  in  diameter,  and  about  a  yard  in  length, 
is  completely  filled  with  mercury;  the  extremity  is  then  stopped 
with  the  finger,  and  the  tube  is  inverted  in  a  vessel  containing 
mercury.     If  the  finger  is  now  removed,  the  mercury  will  descend 
in  the  tube,  and  after  a  few  oscillations  will  remain  stationary  at  a 
height  which  varies  according  to  circumstances,  but  which  is  gen- 
erally about  76  centimetres,  or  nearly  30  inches.1 

The  column  of  mercury  is  maintained  at  this  height  by  the  pres- 
sure of  the  atmosphere  upon  the  surface  of  the  mercury 
in  the  vessel..  In  fact,  the  pressure  at  the  level  ABCD 
(Fig.  106)  must  be  the  same  within  as  without  the  tube; 
so  that  the  column  of  mercury  BE  exerts  a  pressure  equal 
to  that  of  the  atmosphere. 

Accordingly,  we  conclude  from  this  experiment  of 
Torricelli  that  every  surface  exposed  to  the  atmosphere 
sustains  a  normal  pressure  equal,  on  an  average,  to  the 
weight  of  a  column  of  mercury  whose  base  is  this  surface, 
and  whose  height  is  30  inches. 

It  is  evident  that  if  we  performed  a  similar  experi- 
ment with  water,  whose  density  is  to  that  of  mercury  as 
1  :  13'59,  the  height  of  the  column  sustained  would  be 
13-59  times  as  much;  that  is,  30xl3'59  inches,  or  about 
34  feet.  This  is  the  maximum  height  to  which  water 
can  be  raised  in  a  pump;  as  was  observed  by  Galileo. 

In  general  the  heights  of  columns  of  different  liquids 
equal  in  weight  to  a  column  of  air  on  the  same  base,  are 
inversely  proportional  to  their  densities. 

198.  Pressure  of  one  Atmosphere. — What  is  usually  adopted  in 
accurate  physical  discussions  as  the  standard  "  atmosphere  "  of  pres- 
sure is  the  pressure  due  to  a  height  of  76  centimetres  of  pure  mercury 
at  the  temperature  zero  Centigrade,  gravity  being  supposed  to  have 

1  76  centimetres  are  29'922  inches. 


Fig.  106. 


142  THE  BAROMETER. 

the  same  intensity  which  it  has  at  Paris.  The  density  of  mercury  at 
this  temperature  is  13'596;  hence,  when  expressed  in  gravitation 
measure,  this  pressure  is  76  X  13-596  =  1033'3  grammes  per  square 
centimetre.1  To  reduce  this  to  absolute  measure,  we  must  multiply 
by  the  value  of  g  (the  intensity  of  gravity)  at  Paris,  which  is  980-94; 
and  the  result  is  1013600,  which  is  the  intensity  of  pressure  in 
dynes  per  square  centimetre.  In  some  recent  works,  the  round 
number  a  million  dynes  per  square  centimetre  has  been  adopted  as 
the  standard  atmosphere. 

199.  Pascal's  Experiments. — It  is  supposed,  though  without  any 
decisive  proof,  that  Torricelli  derived  from   Galileo   the   definite 
conception  of  atmospheric  pressure.2     However  this  may  be,  when 
the  experiment  of  the  Italian  philosopher  became  known  in  France 
in  1644,  no  one  was  capable  of  giving  the  correct  explanation  of  it, 
and  the  famous  doctrine  that  "  nature  abhors  a  vacuum,"  by  which 
the  rising  of  water  in  a  pump  was  accounted  for,  was  generally 
accepted.     Pascal  was  the  first  to  prove  incontestably  the  falsity  of 
this  old  doctrine,  and  to  introduce  a  more  rational  belief.     For  this 
purpose,  he  proposed  or  executed  a  series  of  ingenious  experiments, 
and  discussed  minutely  all  the  phenomena  which  were  attributed  to 
nature's  abhorrence  of  a  vacuum,  showing  that  they  were  necessary 
consequences  of  the  pressure  of  the  atmosphere. 

We  may  cite  in  particular  the  observation,  made  at  his  suggestion, 
that  the  height  of  the  mercurial  column  decreases  in  proportion  as 
we  ascend.  This  beautiful  and  decisive  experiment,  which  is  repeated 
as  often  as  heights  are  measured  by  the  barometer,  and  which  leaves 
no  doubt  as  to  the  nature  of  the  force  which  sustains  the  mercurial 
column,  was  performed  for  the  first  time  at  Clermont,  and  on  the 
top  of  the  mountain  Puy-de-D6me,  on  the  19th  September,  1648. 

200.  The  Barometer. — By  fixing  the  Torricellian  tube  in  a  perman- 

1  This  is  about  147  pounds  per  square  inch. 

2  In  the  fountains  of  the  Grand-duke  of  Tuscany  some  pumps  were  required  to  raise 
water  from  a  depth  of  from  40  to  50  feet.     When  these  were  worked,  it  was  found  that 
they  would  not  draw.     Galileo  determined  the  height  to  which  the  water  rose  in  their 
tubes,  and  found  it  to  be  about  32  feet;  and  as  he  had  observed  and  proved  that  air  has 
weight,  he  readily  conceived  that  it  was  the  weight  of  a  column  of  the  atmosphere  which 
maintained  the  water  at  this  height  in  the  pumps.     No  very  useful  results,  however,  were 
expected  from  this  discovery,  until,  at  a  later  date,  Torricelli  adopted  and  greatly  extended 
it.     Desiring  to  repeat  the  experiment  in  a  more  convenient  form,  he  conceived  the  idea 
of  substituting  for  water  a  liquid  that  is  14  times  as  heavy,  namely,  mercury,  rightly 
imagining  that  a  column  of  one-fourteenth  of  the  length  would  balance  the  force  which 
sustained  32  feet  of  water  (Biot,  Biographic  UniverseJlc,  article  "  Torricelli ").— 2>. 


CISTERN   BAROMETER. 


143 


ent  position,  we  obtain  a  means  of  measuring  the  amount  of  the 
atmospheric  pressure  at  any  moment;  and  this  pressure  may  be  ex- 
pressed by  the  height  of  the  column  of  mercury  which  it  supports. 
Such  an  instrument  is  called  a  barometer.  In  order  that  its  indica- 
tions may  be  accurate,  several  precautions  must  be  observed.  In  the 
first  place,  the  liquid  used  in  different  barometers 
must  be  identical;  for  the  height  of  the  column 
supported  naturally  depends  upon  the  density  of 
the  liquid  employed,  and  if  this  varies,  the  obser- 
vations made  with  different  instruments  will  not 
be  comparable. 

The  mercury  employed  is  chemically  pure, 
being  generally  made  so  by  washing  with  a  dilute 
acid  and  by  subsequent  distillation.  The  baro- 
metric tube  is  filled  nearly  full,  and  is  then  placed 
upon  a  sloping  furnace,  and  heated  till  the  mer- 
cury boils.  The  object  of  this  process  is  to  expel 
the  air  and  moisture  which  may  be  contained  in 
the  mercurial  column,  and  which,  without  this  pre- 
caution, would  gradually  ascend  into  the  vacuum 
above,  and  cause  a  downward  pressure  of  un- 
certain amount,  which  would  prevent  the  mercury 
from  rising  to  the  proper  height. 

The  next  step  is  to  fill  up  the  tube  with  pure 
mercury,  taking  care  not  to  introduce  any  bubble 
of  air.  The  tube  is  then  inverted  in  a  cistern 
likewise  containing  pure  mercury  recently  boiled, 
and  is  firmly  fixed  in  a  vertical  position,  as  shown 
in  Fig.  107. 

We  have  thus  a  fixed  barometer;  and  in  order 
to  ascertain  the  atmospheric  pressure  at  any 
moment,  it  is  only  necessary  to  measure  the 
height  of  the  top  of  the  column  of  mercury  above 
the  surface  of  the  mercury  in  the  cistern.  One 
method  of  doing  this  is  to  employ  an  iron  rod, 
working  in  a  screw,  and  fixed  vertically  above  the 
surface  of  the  mercury  in  the  dish.  The  extremities  of  this  rod  are 
pointed,  and  the  lower  extremity  being  brought  down  to  touch  the 
surface  of  the  liquid  below,  the  distance  of  the  upper  extremity  from 
the  top  of  the  column  of  mercury  is  measured.  Adding  to  this  the 


Fig.  107. — Barometer  : 
its  simplest  form. 


144  THE   BAROMETER. 

length  of  the  rod,  which  has  previously  been  determined  once  for 
all,  we  have  the  barometric  height.  This  measurement  may  be 
effected  with  great  precision  by  means  of  the  cathetometer. 

201.  Cathetometer.— This  instrument,  which  is  so  frequently  em- 

ployed in  physics  to  measure  the 
vertical  distance  between  two  points, 
was  invented  by  Dulong  and  Petit. 
It  consists  essentially  (Fig.  108)  of 
a  vertical  scale  divided  usually  into 
half  millimetres.  This  scale  forms 
part  of  a  brass  cylinder  capable  of 
turning  very  easily  about  a  strong 
steel  axis.  This  axis  is  fixed  on  a 
pedestal  provided  with  three  levelling 
screws,  and  with  two  spirit-levels  at 
right  angles  to  each  other.  Along 
the  scale  moves  a  sliding  frame  carry- 
ing a  telescope  furnished  with  cross- 
wires,  that  is,  with  two  very  fine 
threads,  usually  spider  lines,  in  the 
focus  of  the  eye-piece,  whose  point  of 
intersection  serves  to  determine  the 
line  of  vision.  By  means  of  a  clamp 
and  slow-motion  screw,  the  telescope 
can  be  fixed  with  great  precision  at 
any  required  height.  The  telescope 
is  also  provided  with  a  spirit-level 
and  adjusting  screw.  When  the 
apparatus  is  in  correct  adjustment, 
the  line  of  vision  of  the  telescope  is 
horizontal,  and  the  graduated  scale  is 
vertical.  If  then  we  wish  to  measure 
the  difference  of  level  between  two 
Fig.  103. -cathetometer.  points,  we  have  only  to  sight  them 

successively,  and  measure  the  distance 

passed  over  on  the  scale,  which  is  done  by  means  of  a  vernier 

attached  to  the  sliding  frame. 

202.  Fortin's  Barometer.— The  barometer  just  described  is  intended 
to  be  fixed;  when  portability  is  required,  the  construction  devised  by 
Fortin  (Fig.  109)  is  usually  employed.      It  is  also  frequently  em- 


CATHETOMETER. 


145 


ployed  for  fixed  barometers.  The  cistern,  which  is  formed  of  a  tube 
of  boxwood,  surmounted  by  a  tube  of  glass,  is  closed  below  by  a 
piece  of  leather,  which  can  be  raised  or  lowered  by  means  of  a  screw. 
This  screw  works  in  the  bottom  of  a  brass  case,  which  incloses  the 
cistern  except  at  the  middle,  where  it  is  cut 
away  in  front  and  at  the  back,  so  as  to  leave 
the  surface  of  the  mercury  open  to  view.  The 
barometric  tube  is  encased  in  a  tube  of  brass 
with  two  slits  at  opposite  sides  (Fig.  110);  and 
it  is  on  this  tube  that  the  divisions  are  engraved, 
the  zero  point  from  which  they  are  reckoned 
being  the  lower  extremity  of  an  ivory  point 
fixed  in  the  covering  of  the  cistern.  The  tem- 
perature of  the  mercury,  which  is  required  for 
one  of  the  corrections  mentioned  in  next  section, 
is  given  by  a  thermometer  with  its  bulb  resting 
against  the  tube.  A  cylindrical 
sliding  piece  (shown  in  Fig.  110) 
furnished  with  a  vernier,1  moves 
along  the  tube  and  enables  us  to 
determine  the  height  with  great 
precision.  Its  lower  edge  is  the 
zero  of  the  vernier.  The  way  in 
which  the  barometric  tube  is 
fixed  upon  the  cistern  is  worth 
notice.  In  the  centre  of  the 
upper  surface  of  the  copper  casing 
there  is  an  opening,  from  which 
rises  a  short  tube  of  the  same 
metal,  lined  with  a  tube  of  box- 
wood. The  barometric  tube  is 
pushed  inside,  and  fitted  in  with 
a  piece  of  chamois  leather,  which 
prevents  the  mercury  from  issuing,  but  does  not  exclude  the  air, 
which,  passing  through  the  pores  of  the  leather,  penetrates  into  the 
cistern,  and  so  transmits  its  pressure. 

Before  taking  an  observation,  the  surface  of  the  mercury  is  ad- 

1  The  vernier  is  an  instrument  very  largely  employed  for  measuring  the  fractions  of  a 
unit  of  length  on  any  scale.     Suppose  we  have  a  scale  divided  into  inches,  and  another 
scale  containing  nine  inches  divided  into  ten  equal  parts.    If  now  we  make  the  end  of  this 
10 


Fig.  110. 

Upper  portion  of 
Barometer. 


Fig.  109. 

Cistern  of  Fortin's 
Barometer. 


•L4G  THE  BAROMETER. 

justed,  by  means  of  the  lower  screw,  to  touch  the  ivory  point.  The 
observer  knows  when  this  condition  is  fulfilled  by  seeing  the 
extremity  of  the  point  touch  its  image  in  the  mercury.  Tbe  sliding 
piece  which  carries  the  vernier  is  then  raised  or  lowered,  until  its 
base  is  seen  to  be  tangential  to  the  upper  surface  of  the  mercurial 
column,  as  shown  in  Fig.  110.  In  making  this  adjustment,  the  back 
of  the  instrument  should  be  turned  towards  a  good  light,  in  order 
that  the  observer  may  be  certain  of  the  position  in  which  the  light 
is  just  cut  off  at  the  summit  of  the  convexity. 

When  the  instrument  is  to  be  carried  from  place  to  place,  precau- 
tions must  be  taken  to  prevent  the  mercury  from  bumping  against 
the  top  of  the  tube  and  breaking  it.  The  screw  at  the  bottom  is  to 
be  turned  until  the  mercury  reaches  the  top  of  the  tube,  and  the 
instrument  is  then  to  be  inverted  and  carried  upside  down. 

We  may  here  remark  that  the  goodness  of  the  vacuum  in  a  bar- 
ometer, can  be  tested  by  the  sound  of  the  mercury  when  it  strikes 
the  top  of  the  tube,  which  it  can  be  made  to  do  either  by  screwing 

Utter  scale,  which  is  called  the  vernier,  coincide  with  one  of  the  divisions  in  the  scale  of 
inches,  as  each  division  of  the  vernier  is  T97  of  an  inch,  it  is  evident  that  the  first  division 
on  the  scale  will  be  ^  of  an  inch  beyond  the  first  division  on  the  vernier,  the  second  on 
the  scale  TST  beyond  the  second  on  the  vernier,  and  so  on  until  the  ninth  on  the  scale,  which 


',  ',  \  \   1    1   *   ' 

•  •  f, 

PTTT3 

n  ,  .  .  • .  .  ,T 


.u. 


Fig.  111.— Vernier. 

will  exactly  coincide  with  the  tenth  on  the  vernier.  Suppose  next  that  in  measuring  any 
length  we  find  that  its  extremity  lies  between  the  degrees  5  and  6  on  the  scale;  we  bring 
the  zero  of  the  vernier  opposite  the  extremity  of  the  length  to  be  measured,  and  observe 
what  division  on  the  vernier  coincides  with  one  of  the  divisions  on  the  scale.  We  see  in 
the  figure  that  it  is  the  seventh,  and  thus  we  conclude  that  the  fraction  required  is  T7^  of 
an  inch.  , 

If  the  vernier  consisted  of  19  inches  divided  into  20  equal  parts,  it  would  read  to  the  ^ 
of  an  inch ;  but  there  is  a  limit  to  the  precision  that  can  thus  be  obtained.  An  exact  coin- 
cidence of  a  division  on  the  vernier  with  one  on  the  scale  seldom  or  never  takes  place,  and 
we  merely  take  the  division  which  approaches  nearest  to  this  coincidence;  so  that  when 
the  difference  between  the  degrees  on  the  vernier  and  those  on  the  scale  is  very  small, 
there  may  be  so  much  uncertainty  in  this  selection  as  to  nullify  the  theoretical  precision 
of  the  instrument.  Verniers  are  also  employed  to  measure  angles ;  when  a  circle  is  divided 
into  half  degrees,  a  vernier  is  used  which  gives  -fa  of  a  division  on  the  circle,  that  is,  -^ 
of  a  half  degree,  or  one  minute. — D. 


PORTABLE   BAROMETER. 


147 


up  or  by  inclining  the  instrument  to  one  side.  If  the  vacuum  is 
good,  a  metallic  clink  will  be  heard,  and  unless  the  contact  be  made 
very  gently,  the  tube 
will  be  broken  by  the 
sharpness  of  the'  col- 
lision. If  any  air  be 
present,  it  acts  as  a 
cushion. 

In  making  observa- 
tions in  the  field,  a 
barometer  is  usually 
suspended  from  a  tri- 
pod stand  (Fig.  112) 
by  gimbals1,  so  that  it 
always  takes  a  vertical 
position. 

203.  Float  Adjust- 
ment.— In  some  barom- 
eters the  ivory  point  for 
indicating    the    proper 
level  of  the  mercury  in 
the  cistern  is  replaced 
by  a  float.    F  (Fig.  113) 
is  a  small  ivory  piston, 
having    the    float    at- 
tached to  its  foot,  and 
moving  freely  up  and 
down  between  the  two 
ivory  guides  I.    A  hori- 
zontal line  (interrupted 
by   the   piston)    is   en- 
graved    on     the     two 
guides,  and  another  is 
engraved  on  the  piston, 

at  such  a  height  that  the  three  lines  form  one  straight  line  when  the 
surface  of  the  mercury  in  the  cistern  stands  at  the  zero  point  of  the 
scale. 

204.  Barometric  Corrections. — In  order  that  barometric  heights 

1  A  kind  of  universal  joint,  in  common  use  on  board  ship  for  the  suspension  of  com- 
passes, lamps,  &c.     It  is  seen  in  Fig.  112,  at  the  top  of  the  tripod  stand. 


Fig.  112. — Barometer  "with  Tiipod  Stand. 


148 


THE   BAROMETER. 


may  be  comparable  as  measures  of  atmospheric  pressure,  certain  cor- 
rections must  be  applied. 

1.  Correction  for  Temperature.  As  mercury  expands  with  heat, 
it  follows  that  a  column  of  warm  mercury  exerts  less 
pressure  than  a  column  of  the  same  height  at  a  lower 
temperature;  and  it  is  usual  to  reduce  the  actual 
height  of  the  column  to  the  height  of  a  column  at  the 
temperature  of  freezing  water  which  would  exert  the 
same  pressure. 

Let  h  be  the  observed  height  at  temperature  t° 
Centigrade,  and  h0  the  height  reduced  to  freezing- 
point.  Then,  if  m  be  the  coefficient  of  expansion  of 
mercury  per  degree  Cent.,  we  have 

A0  (1  +m  t)  =h,  whence  ho  =  h-hmt  nearly. 

The  value  of  m  is  -^='00018018.    For  temperatures 
Fahrenheit,  we  have 


Fioatl&ent.    where  m  denotes 


=  -0001001. 
But    temperature    also  affects  the 


length   of  the 

divisions  on  the  scale  by  which  the  height  of  the  mercurial  column 
is  measured.  If  these  divisions  be  true  inches  at  0°  Cent.,  then  at 
f  the  length  of  n  divisions  will  be  n  (1  +  It)  inches,  I  denoting  the 
coefficient  of  linear  expansion  of  the  scale,  the  value  of  which  for 
brass,  the  usual  material,  is  '00001878.  If  then  the  observed  height 
h  amounts  to  n  divisions  of  the  scale,  we  have 


whence 


1  +  mt 


that  is  to  say,  if  n  be  the  height  read  off  on  the  scale,  it  must  be 
diminished  by  the  correction  n  t  (m  —  l),  t  denoting  the  temperature 
of  the  mercury  in  degrees  Centigrade.  The  value  of  m  —  l  is 
•0001614. 

For  temperatures  Fahrenheit,  assuming  the  scale  to  be  of  the 
correct  length  at  32°  Fahr.,  the  formula  for  the  correction  (which  is 
still  subtractive),  is  n  (<—  32)  (m—l),  where  m—  I  has  the  value 
•00008967.1 

1  The  correction  for  temperature  is  usually  made  by  the  help  of  tables,  which  give  its 
amount  for  all  ordinary  temperatures  and  heights.  These  tables,  when  intended  for 


CORKECTIONS.  149 

2.  Correction  for  Capillarity.  —  In  the  preceding  chapter  we  have 
seen  that  mercury  in  a  glass  tube  undergoes  a  capillary  depression: 
whence  it  follows  that  the  observed  barometric  height  is  too  small, 
and  that  we  must  add  to  it  the  amount  of  this  depression.  In  all 
tubes  of  internal  diameter  less  than  about  f  of  an  inch  this  correction 
is  sensible;  and  its  amount,  for  which  no  simple  formula  can  be  given, 
has  been  computed,  from  theoretical  considerations,  for  various  sizes 
of  tube,  by  several  eminent  mathematicians,  and  recorded  in  tables, 
from  which  that  given  below  is  abridged.  These  values  are  appli- 
cable on  the  assumption  that  the  meniscus  which  forms  the  summit 
of  the  mercurial  column  is  decidedly  convex,  as  it  always  is  when 
the  mercury  is  rising.  When,  the  meniscus  is  too  flat,  the  mercury 
must  be  lowered  by  the  foot-screw,  and  then  screwed  up  again. 

It  is  found  by  experiment,  that  the  amount  of  capillary  depression 
is  only  half  as  great  when  the  mercury  has  been  boiled  in  the  tube 
as  when  this  precaution  has  been  neglected. 

For  purposes  of  special  accuracy,  tables  have  been  computed, 
giving  the  amount  of  capillary  depression  for  different  degrees  of 
convexity,  as  determined  by  the  sagitta  (or  height)  of  the  meniscus, 
taken  in  conjunction  with  the  diameter  of  the  tube.  Such  tables, 
however,  are  seldom  used  in  this  country.1 

English  barometers,  are  generally  constructed  on  the  assumption  that  the  scale  is  of  the 
correct  length  not  at  32°  Fahr.,  but  at  62J  Fahr.,  which  is  (by  act  of  Parliament)  the 
temperature  at  which  the  British  standard  yard  (preserved  in  the  office  of  the  Exchequer) 
is  correct.  On  this  supposition,  the  length  of  n  divisions  of  the  scale  at  temperature  ta 
Fahr.,  is 

»{!  +  *(*  -62)}; 
and  by  equating  this  expression  to 

h0{l  +  m  (t  -  31)} 
we  find 

A0  =  «{l  -m  (*-32)+Z(«-62)} 

=  n|  1  -  (m-l)t+  (32m  -  62Z)  ]• 
=  K  1  1  -  -00008967  t  +  -00255654  j-  ; 
which,  omitting  superfluous  decimals,  may  conveniently  be  put  in  the  fi  nn  — 


The  correction  vanishes  when 

•09  t  -  2-56  =  0; 

that  is,  when  t=~  =  28-5. 

For  all  temperatures  higher  than  this  the  correction  is  subtractive. 

1  The  most  complete  collection  of  meteorological  and  physical  tables,  is  that  edited  by 
Professor  Guyot,  and  published  under  the  auspices  of  the  Smithsonian  Institution,  Wash- 
ington. 


150 


THE   BAROMETER. 


TABLE  OF  CAPILLARY  DEPRESSIONS  IN  UNBOILED  TUBES. 
(To  be  halved  for  Soiled  Tubes.) 


Diameter  of 
tube  ill  inches. 

Depression  in 
inches. 

Diameter. 

Depression. 

Diameter. 

Depression. 

•10 

•140 

•20 

•058 

•40 

•015 

•11 

•126 

•22 

•050 

•42 

•013 

•12 

•114 

•24 

•044 

•44 

•Oil 

•13 

•104 

•26 

•038 

•46 

•009 

•14 

•094 

•28 

•033 

•48 

•008 

•15 

•086 

•30 

•029 

•50 

•007 

•16 

•079 

•32 

•026 

•55 

•005 

•17 

•073 

•34 

•023 

•60 

•004 

•18 

•068 

•36 

•020 

•65 

•003 

•19 

•063 

•38 

•017 

•70 

•002 

3.  Correction  for  Capacity.  —  When  there  is  no  provision  for  ad- 
justing the  level  of  the  mercury  in  the  cistern  to  the  zero  point  of 
the  scale,  another  correction  must  be  applied.  It  is  called  the  cor- 
rection for  capacity.  In  barometers  of  this  construction,  which  were 
formerly  much  more  common  than  they  are  at  present,  there  is  a 
certain  point  in  the  scale  at  which  the  mercurial  column  stands  when 
the  mercury  in  the  cistern  is  at  the  correct  level.  This  is  called  the 
neutral  point.  If  A  be  the  interior  area  of  the  tube,  and  C  the  area 
of  the  cistern  (exclusive  of  the  space  occupied  by  the  tube  and  its 
contents),  when  the  mercury  in  the  tube  rises  by  the  amount  x,  the 

mercury  in  the  cistern  falls  by  an  amount  y  =  ~x;  for  the  volume  of 

the  mercury  which  has  passed  from  the  cistern  into  the  tube  is 
C  y  =  A  x.  The  change  of  atmospheric  pressure  is  correctly  measured 
l-|-«  an(i  ^  we  now  take  x  to  denote  the  distance  of 


the  summit  of  the  mercurial  column  from  the  neutral  point,  the  cor- 
rected distance  will  be  (  ~L  +  ^\x,  and  the  correction  to  be  applied  to 

the  observed  reading  will  be  ^  x,  which  is  additive  if  the  observed 
reading  be  above  the  neutral  point,  subtractive  if  below. 

It  is  worthy  of  remark  that  the  neutral  point  depends  upon  the 
volume  of  mercury.  It  will  be  altered  if  any  mercury  be  lost  or 
added;  and  as  temperature  affects  the  volume,  a  special  temperature- 
correction  must  be  applied  to  barometers  of  this  class.  The  investi- 
gation will  be  found  in  a  paper  by  Professor  Swan  in  the  Philo- 
sophical Magazine  for  1861. 

In  some  modern  instruments  the  correction  for  capacity  is  avoided, 
by  making  the  divisions  on  the  scale  less  than  true  inches,  in  the 


CORRECTIONS.  151 

ratio  -£ — £,  and  the  effect  of  capillarity  is  at  the  same  time  compen- 
sated by  lowering  the  zero  point  of  the  scale.  Such  instruments,  if 
correctly  made,  simply  require  to  be  corrected  for  temperature. 

4.  Index  Errors. — Under  this  name  are  included  errors  of  gradua- 
tion, and  errors  in  the  position  of  the  zero  of  the  graduations.     An 
error  of  zero  makes  all  readings  too  high  or  too  low  by  the  same 
amount.     Errors  of  graduation  (which  are  generally  exceedingly 
small)  are  different  for  different  parts  of  the  scale. 

Barometers  intended  for  accurate  observation  are  now  usually 
examined  at  Kew  Observatory  before  being  sent  out;  and  a  table  is 
furnished  with  each,  showing  its  index  error  at  every  half  inch  of 
the  scale,  errors  of  capillarity  and  capacity  (if  any)  being  included 
as  part  of  the  index  error.  We  may  make  a  remark  here  once  for 
all  respecting  the  signs  attached  to  errors  and  corrections.  The 
sign  of  an  error  is  always  opposite  to  that  of  its  correction.  When 
a  reading  is  too  high  the  index  error  is  one  of  excess,  and  is  there- 
fore positive;  whereas  the  correction  needed  to  make  the  reading 
true  is  subtractive,  and  is  therefore  negative. 

5.  Reduction  to  Sea-level. — In  comparing  barometric  observations 
taken  over  an  extensive  district  for  meteorological  purposes,  it  is 
usual  to  apply  a  correction  for  difference  of  level.     Atmospheric 
pressure,  as  we  have  seen,  diminishes  as  we  ascend;  and  it  is  usual 
to  add  to  the  observed  height  the  difference  of  pressure  due  to  the 
elevation  of  the  place  above  sea-level.     The  amount  of  this  correc- 
tion is  proportional  to  the  observed  pressure.     The  law  according  to 
which  it  increases  with  the  height  will  be  discussed  in  the  next 
chapter. 

6.  Correction  for    Unequal  Intensity   of  Gravity. — When   two 
barometers  indicate  the  same  height,  at  places  where  the  intensity 
of  gravity  is  different  (for  example,  at  the  pole  and  the  equator), 
the  same  mass  of  air  is  superincumbent  over  both;  but  the  pressures 
are  unequal,   being  proportional   to   the   intensity   of  gravity   as 
measured  by  the  values  of  g  (§  91)  at  the  two  places. 

If  h  be  the  height,  in  centimetres,  of  the  mercurial  column  at  the 
temperature  0°  Cent.,  the  absolute  pressure,  in  dynes  per  square 
centimetre,  will  be  gh  X  13-596;  since  13'59G  is  the  density  of 
mercury  at  this  temperature. 

205.  Other  kinds  of  Mercurial  Barometer. — The  Siphon  Barometer, 
which  is  represented  in  Fig.  114,  consists  of  a  bent  tube,  generally 


152 


THE   BAROMETER. 


of  uniform  bore,  having  two  unequal  legs.  The  longer  leg,  which 
must  be  more  than  30  inches  long,  is  closed,  while  the  shorter  leg  is 
open.  A  sufficient  quantity  of  mercury  having  been  introduced  to 
fill  the  longer  leg,  the  instrument  is  set  upright  (after  boiling  to 
expel  air),  and  the  mercury  takes  such  a  position  that  the 
difference  of  levels  in  the  two  legs  represents  the  pressure 
of  the  atmosphere. 

Supposing  the  tube  to  be  of  uniform  section,  the  mer- 
cury will  always  fall  as  much  in  one  leg  as  it  rises  in  the 
other.  Each  end  of  the  mercurial  column  therefore  rises 
or  falls  through  only  half  the  height  corresponding  to 
the  change  of  atmospheric  pressure. 

In  the  best  siphon  barometers  there  are  two  scales,  one 
for  each  leg,  as  indicated  in  the  figure,  the  divisions  on 
one  being  reckoned  upwards,  and  on  the  other  down- 
wards, from  an  intermediate  zero  point,  so  that  the  sum 
of  the  two  readings  is  the  difference  of  levels  of  the 
mercury  in  the  two  branches. 

Inasmuch  as  capillarity  tends  to  depress  both  extrem- 
ities of  the  mercurial  column,  its  effect  is  generally 
neglected  in  siphon  barometers;  but  practically  it  causes 
great  difficulty  in  obtaining  accurate  observations,  for 
according  as  the  mercury  is  rising  or  falling  its  ex- 
tremity is  more  or  less  convex,  and  a  great  deal  of  tapping  is 
usually  required  to  make  both  ends  of  the  column  assume  the  same 
form,  which  is  the  condition  necessary  for  annihilating  the  effect  of 
capillary  action. 

Wheel  Barometer. — The  wheel  barometer,  which  is  in  more  gen- 
eral use  than  its  merits  deserve,  consists  of  a  siphon  barometer, 
the  two  branches  of  which  have  usually  the  same  diameter.  On 
the  surface  of  the  mercury  of  the  open  branch  floats  a  small  piece 
of  iron  or  glass  suspended  by  a  thread,  the  other  extremity  of  which 
is  fixed  to  a  pulley,  on  which  the  thread  is  partly  rolled.  Another 
thread,  rolled  parallel  to  the  first,  supports  a  weight  which  balances 
the  float.  To  the  axis  of  the  pulley  is  fixed  a  needle  which  moves  on 
a  dial.  When  the  level  of  the  mercury  varies  in  either  direction, 
the  float  follows  its  movement  through  the  same  distance;  by  the 
action  of  the  counterpoise  the  pulley  turns,  and  with  it  the  needle, 
the  extremity  of  which  points  to  the  figures  on  the  dial,  marking 
the  barometric  heights.  The  mounting  of  the  dial  is  usually  placed 


Fig.  114. 

Siphon 

Barometer. 


SIPHON   AND   WHEEL   BAROMETERS. 


153 


in  front  of  the  tube,  so  as  to  conceal  its  presence.  The  wheel 
barometer  is  a  very  old  invention,  and  was  introduced  by  the 
celebrated  Hooke  in  1683.  The  pulley  and  strings  are  sometimes 
replaced  by  a  rack  and  pinion,  as  represented  in  the  figure 
(Fig.  115). 

Besides  the  faults  incidental  to  the  siphon  barometer,  the  wheel 


Pig.  115.— Wheel  Barometer. 


barometer  is  encumbered  in  its  movements  by  the  friction  of  the 
additional  apparatus.  It  is  quite  unsuitable  for  measuring  the 
exact  amount  of  atmospheric  pressure,  and  is  slow  in  indicating 
changes. 

Marine  Barometer, — The  ordinary  mercurial  barometer  cannot  be 
used  at  sea  on  account  of  the  violent  oscillations  which  the  mercury 
would  experience  from  the  motion  of  the  vessel.  In  order  to  meet 
this  difficulty,  the  tube  is  contracted  in  its  middle  portion  nearly  to 


154 


THE   BAROMETER. 


capillary  dimensions,  so  that  the  motion  of  the  mercury  in  either 
direction  is  hindered.  An  instrument  thus  constructed  is  called  a 
marine  barometer.  When  such  an  instrument  is  used  on  land  it 
is  always  too  slow  in  its  indications. 

206.  Aneroid  Barometer  (a,  rqpoc).— This  barometer  depends  upon 
the  changes  in  the  form  of  a  thin  metallic  vessel  partially  exhausted 


Fig.  116.— Aueroid  Barometer. 

of  air,  as  the  atmospheric  pressure  varies.  M.  Vidie  was  the  first  to 
overcome  the  numerous  difficulties  which  were  presented  in  the  con- 
struction of  these  instruments.  We  subjoin  a  figure  of  the  model 
which  he  finally  adopted. 

The  essential  part  is  a  cylindrical  box  partially  exhausted  of  air, 
the  upper  surface  of  which  is  corrugated  in  order  to  make  it  yield 
more  easily  to  external  pressure.  At  the  centre  of  the  top  of  the 
box  is  a  small  metallic  pillar  M,  connected  with  a  powerful  steel 
spring  R.  As  the  pressure  varies,  the  top  of  the  box  rises  or  falls, 
transmitting  its  movement  by  two  levers  I  and  m,  to  a  metallic  axis 
r.  This  latter  carries  a  third  lever  t,  the  extremity  of  which  is 
attached  to  a  chain  s  which  turns  a  drum,  the  axis  of  which  bears 
the  index  needle.  A  spiral  spring  keeps  the  chain  constantly 
stretched,  and  thus  makes  the  needle  always  take  a  position  corre- 


ANEROID.  155 

spending  to  the  shape  of  the  box  at  the  time.  The  graduation  is 
performed  empirically  by  comparison  with  a  mercurial  barometer. 
The  aneroid  barometer  is  very  quick  in  indicating  changes,  and  is 
much  more  portable  than  any  form  of  mercurial  barometer,  being 
both  lighter  and  less  liable  to  injury.  It  is  sometimes  made  small 
enough  for  the  waistcoat  pocket.  It  has  the  drawback  of  being 
affected  by  temperature  to  an  extent  which  must  be  determined  for 
each  instrument  separately,  and  of  being  liable  to  gradual  changes 
which  can  only  be  checked  by  occasional  comparison  with  a  good 
mercurial  barometer. 

In  the  metallic  barometer,  which  is  a  modification  of  the  aneroid, 
the  exhausted  box  is  crescent-shaped,  and  the  horns  of  the  crescent 
separate  or  approach  according  as  the  external  pressure  diminishes 
or  increases. 

207.  Old  Forms  Revived. — There  are  two  ingenious  modifications 
of  the  form  of  the  barometer,  which,  after  long  neglect,  have  recently 
been  revived  for  special  purposes. 

Counterpoised  Barometer. — The  invention  of  this  instrument  is 
attributed  to  Samuel  Morland,  who  constructed  it  about  the  year 
1680.  It  depends  upon  the  following  principle: — If  the  barometric 
tube  is  suspended  from  one  of  the  scales  of  a  balance,  there  will  be 
required  to  balance  it  in  the  other  scale  a  weight  equal  to  the  weight 
of  the  tube  and  the  mercury  contained  in  it,  minus  the  upward 
pressure  due  to  the  liquid  displaced  in  the  cistern.1  If  the  atmo- 
spheric pressure  increases,  the  mercury  will  rise  in  the  tube,  and 
consequently  the  weight  of  the  floating  body  will  increase,  while 
the  sinking  of  the  mercury  in  the  cistern  will  diminish  the  upward 
pressure  due  to  the  displacement.  The  beam  will  thus  incline  to 

1  A  complete  investigation  based  on  the  assumption  of  a  constant  upward  pull  at  the 
top  of  the  suspended  tube  shows  that  the  sensitiveness  of  the  instrument  depends  only  on 
the  internal  section  of  the  upper  part  of  the  tube  and  the  external  section  of  its  lower 
part.  Calling  the  former  A  and  the  latter  B,  it  is  necessary  for  stability  that  B  be 
greater  than  A  (which  is  not  the  case  in  the  figure  in  the  text)  and  the  movement  of  the 
tube  will  be  to  that  of  the  mercury  in  a  standard  barometer  as  A  is  to  B  -  A.  The 
directions  of  these  movements  will  be  opposite.  If  B  -  A  is  very  small  compared  with  A, 
the  instrument  will  be  exceedingly  sensitive;  and  as  B-A  changes  sign,  by  passing 
through  zero,  the  equilibrium  becomes  unstable. 

A  curious  result  of  the  investigation  is  that  the  level  of  the  mercury  in  the  cistern  re- 
mains constant. 

In  the  instrument  represented  in  the  figure,  stability  is  probably  obtained  by  the  weight 
of  the  arm  which  carries  the  pencil. 

In  Xing's  barograph,  B  is  made  greater  than  A  by  fixing  a  hollow  iron  drum  round 
the  lower  end  of  the  tube. 


156 


THE   BAROMETER. 


the  side  of  the  barometric  tube,  and  the  reverse  movement  would 
occur  if  the  pressure  diminished.  For  the  balance  may  be  substi- 
tuted, as  in  Fig.  117,  a  lever  carrying  a  counterpoise;  the  variations 
of  pressure  will  be  indicated  by  the  movements  of  this  lever. 

Such  an  instrument  may  very  well  be  used  as  a  barograph  or  re- 
cording barometer;  for 
this  purpose  we  have 
only  to  attach  to  the 
lever  an  arm  with  a 
pencil,  which  is  con- 
stantly in  contact  with 
a  sheet  of  paper  moved 
uniformly  by  clock-work. 
The  result  will  be  a 
continuous  trace,  whose 
form  corresponds  to  the 
variations  of  pressure. 
It  is  very  easy  to  deter- 
mine, either  by  calcula- 
tion or  by  comparison 
with  a  standard  baro- 
meter, the  pressure  cor- 
responding to  a  given 
position  of  the  pencil  on 
the  paper;  and  thus,  if 
the  paper  is  ruled  with 
twenty-four  equidistant 
lines,  corresponding  to 

the  twenty-four  hours  of  the  day,  we  can  see  at  a  glance  what  was  the 
pressure  at  any  given  time.  An  arrangement  of  this  kind  has  been 
adopted  by  the  Abbe  Secchi  for  the  meteorograph  of  the  observatory 
at  Rome.  The  first  successful  employment  of  this  kind  of  barograph 
appears  to  be  due  to  Mr.  Alfred  King,  a  gas  engineer  of  Liverpool, 
who  invented  and  constructed  such  an  instrument  in  1853,  for  the 
use  of  the  Liverpool  Observatory,  and  subsequently  designed  a  larger 
one,  which  is  still  in  use,  furnishing  a  very  perfect  record,  magnified 
five-and-a-half  times. 

Fahrenheit's  Baromeier. — Fahrenheit's  barometer  consists  of  a  tube 
bent  several  times,  the  lower  portions  of  which  contain  mercury;  the 
upper  portions  are  filled  with  water,  or  any  other  liquid,  usually 


Fig.  117. — Counterpoised  Barometer. 


PHOTOGRAPHIC   REGISTRATION. 


157 


Fig.  118.— Fahrenheit's  Barometer. 


coloured.  It  is  evident  that  the  atmospheric  pressure  is  balanced  by 
the  sum  of  the  differences  of  level  of  the  columns  of  mercury,  dimin- 
ished by  the  sum  of  the  corresponding  differences  for  the  columns 
of  water;  whence  it  follows  that,  by 
employing  a  considerable  number  of 
tubes,  we  may  greatly  reduce  the  height 
of  the  barometric  column.  This  circum- 
stance renders  the  instrument  interesting 
as  a  scientific  curiosity,  but  at  the  same 
time  diminishes  its  sensitiveness,  and 
renders  it  unfit  for  purposes  of  precision. 
It  is  therefore  never  used  for  the 
measurement  of  atmospheric  pressure; 
but  an  instrument  upon  the  same  prin- 
ciple has  recently  been  employed  for  the 
measurement  of  very  high  pressures,  as 
will  be  explained  in  Chap.  xix. 

208.  Photographic  Registration. — Since  the  year  1847  various 
meteorological  instruments  at  the  Royal  Observatory,  Greenwich, 
have  been  made  to  yield  continuous  traces  of  their  indications  by  the 
aid  of  photography,  and  the  method  is  now  generally  employed  at 
meteorological  observatories  in  this  country.  The  Greenwich  system 
is  fully  described  in  the  Greenwich  Magnetical  and  Meteorological 
Observations  for  1847,  pp.  Ixiii.-xc.  (published  in  1849). 

The  general  principle  adopted  for  all  the  instruments  is  the  same. 
The  photographic  paper  is  wrapped  round  a  glass  cylinder,  and  the 
axis  of  the  cylinder  is  made  parallel  to  the  direction  of  the  move- 
ment which  is  to  be  registered.  The  cylinder  is  turned  by  clock- 
work, with  uniform  velocity.  The  spot  of  light  (for  the  magnets 
and  barometer),  or  the  boundary  of  the  line  of  light  (for  the  ther- 
mometers), moves,  with  the  movements  which  are  to  be  registered, 
backwards  and  forwards  in  the  direction  of  the  axis  of  the  cylinder, 
while  the  cylinder  itself  is  turned  round.  Consequently  (as  in 
Morin's  machine,  Chap,  vii.),  when  the  paper  is  unwrapped  from  its 
cylindrical  form,  there  is  traced  upon  it  a  curve  of  which  the  abscissa 
is  proportional  to  the  time,  while  the  ordinate  is  proportional  to  the 
movement  which  is  the  subject  of  measure. 

The  barometer  employed  in  connection  with  this  system  is  a  large 
siphon  barometer,  the  bore  of  the  upper  and  lower  extremities  of  its 
arms  being  about  I'l  inch.  A  glass  float  in  the  quicksilver  of  the 


158  THE   BAROMETER. 

lower  extremity  is  partially  supported  by  a  counterpoise  acting  on  a 
lio-ht  lever  (which  turns  on  delicate  pivots),  so  that  the  wire  support- 
ing the  float  is  constantly  stretched,  leaving  a  definite  part  of  the 
weight  of  the  float  to  be  supported  by  the  quicksilver.  This  lever  is 
lengthened  to  carry  a  vertical  plate  of  opaque  mica  with  a  small  aper- 
ture, whose  distance  from  the  fulcrum  is  eight  times  the  distance  of 
the  point  of  attachment  of  the  float-wire,  and  whose  movement, 
therefore  (§  205),  is  four  times  the  movement  of  the  column  of  a  cis- 
tern barometer.  Through  this  hole  the  light  of  a  lamp,  collected  by 
a  cylindrical  lens,  shines  upon  the  photographic  paper. 

Every  part  of  the  cylinder,  except  that  on  which  the  spot  of  light 
falls,  is  covered  with  a  case  of  blackened  zinc,  having  a  slit  parallel 
to  the  axis  of  the  cylinder;  and  by  means  of  a  second  lamp  shining 
through  a  small  fixed  aperture,  and  a  second  cylindrical  lens,  a  base 
line  is  traced  upon  the  paper,  which  serves  for  reference  in  subsequent 
measurements. 

The  whole  apparatus,  or  any  other  apparatus  which  serves  to  give 
a  continuous  trace  of  barometric  indications,  is  called  a  barograph; 
and  the  names  thermograph,  magnetograph,  anemograph,  &c.,  are 
similarly  applied  to  other  instruments  for  automatic  registration. 
Such  registration  is  now  employed  at  a  great  number  of  observa- 
tories; and  curves  thus  obtained  are  regularly  published  in  the 
Quarterly  Reports  of  the  Meteorological  Office. 


CHAPTER    XVIII. 


VARIATIONS   OF  THE  BAROMETER. 


209.  Measurement  of  Heights  by  the  Barometer. — As  the  height  of 
the  barometric  column  diminishes  when  we  ascend   in   the  atmo- 
sphere, it  is  natural  to  seek  in  this  phenomenon  a  means  of  measuring 
heights.     The  problem  would  be  extremely  simple,  if  the  air  had 
everywhere  the  same  density  as  at  the  surface  of  the  earth.     In 
fact,  the  density  of  the  air  at  sea-level  being  about  10,500  times  less 
than  that  of  mercury,  it  follows  that,  on  the  hypothesis  of  uniform 
density,  the  mercurial  column  would  fall  an  inch  for  every  10,500 
inches,  or  875  feet  that  we  ascend.     This  result,  however,  is  far  from 
being  in  exact  accordance  with  fact,  inasmuch  as  the  density  of  the 
air  diminishes  very  rapidly  as  we  ascend,  on  account  of  its  great 
compressibility. 

210.  Imaginary   Homogeneous   Atmosphere. — If   the   atmosphere 
were  of  uniform  and  constant  density,  its  height  would  be  approxi- 
mately obtained  by  multiplying  30  inches  by  10,500,  which  gives 
26,250  feet,  or  about  5  miles. 

More  accurately,  if  we  denote  by  H  the  height  (in  centimetres)  of 
the  atmosphere  at  a  given  time  and  place,  on  the  assumption  that 
the  density  throughout  is  the  same  as  the  observed  density  D  (in 
grammes  per  cubic  centimetre)  at  the  base,  and  if  we  denote  by  P 
the  observed  pressure  at  the  base  (in  dynes  per  square  centimetre), 
we  must  employ  the  general  formula  for  liquid  pressure  (§  139) 

P  =  g  HD,  which  gives  H  =  ^  (1) 

The  height  H,  computed  on  this  imaginary  assumption,  is  usually 
called  the  height  of  the  homogeneous  atmosphere,  corresponding  to 
the  pressure  P,  density  D,  and  intensity  of  gravity  g.  It  is  some- 
times called  the  pressure-height  The  pressure-height  at  any  point 


1GO  VARIATIONS   OF   THE   BAROMETER. 

in  a  liquid  or  gas  is  the  height  of  a  column  of  fluid,  having  the  same 
density  as  at  the  point,  which  would  produce,  by  its  weight,  the 
actual  pressure  at  the  point.  This  element  frequently  makes  its 
appearance  in  physical  and  engineering  problems. 

The  expression  for  H  contains  P  in  the  numerator  and  D  in  the 
denominator;  and  by  Boyle's  law,  which  we  shall  discuss  in  the  ensu- 
ing chapter,  these  two  elements  vary  in  the  same  proportion,  when 
the  temperature  is  constant.  Hence  H  is  not  affected  by  changes  of 
pressure,  but  has  the  same  value  at  all  points  in  the  air  at  which  the 
temperature  and  the  value  of  g  are  the  same. 

7  211.  Geometric  Law  of  Decrease.— The  change  of  pressure  as  we 
ascend  or  descend  for  a  short  distance  in  the  actual  atmosphere,  is 
sensibly  the  same  as  it  would  be  in  this  imaginary  "  homogeneous 
atmosphere;"  hence  an  ascent  of  1  centimetre  takes  off  g  of  the  total 
pressure,  just  as  an  ascent  of  one  foot  from  the  bottom  of  an  ocean 
00,000  feet  deep  takes  off  aoioo  of  the  pressure. 

Since  H  is  the  same  at  all  heights  in  any  portion  of  the  air  which 
is  at  uniform  temperature,  it  follows  that  in  ascending  by  successive 
steps  of  1  centimetre  in  air  at  uniform  temperature,  each  step  takes 

off  the  same  fraction  g  of  the  current  pressure.  The  pressures  there- 
fore form  a  geometrical  progression  whose  ratio  is  1  —  g.  In  an  at- 
mosphere of  uniform  temperature,  neglecting  the  variation  of  g  with 
height,  the  densities  and  pressures  diminish  in  geometrical  progres- 
sion as  the  heights  increase  in  arithmetical  progression. 

212.  Computation  of  Pressure-height. — For  perfectly  dry  air  at  0° 
Cent.,  we  have  the  data  (§§  195,  198), 

D  =  -0012932  when  P  =  1013600; 

which  give 

g  =  783800000  nearly. 

Taking  g  as  981,  we  have 

H  =  i«a{»o -A**  =  799000  centimetres  nearly. 

This  is  very  nearly  8  kilometres,  or  about  5  miles.  At  the  temper- 
ature t°  Cent.,  we  shall  have 

H  =  799000  (1  +  -00366  t).  '   (2) 

Hence  in  air  at  the  the  temperature  0°  Cent.,  the  pressure  diminishes 
by  1  per  cent,  for  an  ascent  of  about  7990  centimetres  or,  say,  80 
metres.  At  20°  Cent.,  the  number  will  be  86  instead  of  80 


HYPSOMETRIC AL   FORMULA.  161 

213.  Formula   for  determining   Heights  by  the  Barometer. — To 
obtain  an  accurate  rule  for  computing  the  difference  of  levels  of  two 
stations  from  observations  of  the  barometer,  we  must  employ  the 
integral  calculus. 

Denote  height  above  a  fixed  level  by  x,  and  pressure  by  p.  Then 
we  have 

dx          dp 
K  =  -  P'> 

and  if  pl}  p2  are  the  pressures  at  the  heights  xl}  x2,  we  deduce  by  in- 
tegration 

Xi  -  Xt  =  H  (loge  Pi  -  loge  p.2). 

Adopting  the  value  of  H  from  (2),  and  remembering  that  Napierian 
logarithms  are  equal  to  common  logarithms  multiplied  by  2*3026,  we 
finally  obtain 

x*  -  *!  =  1840000  (1  +  -00366  t)  (log  Pl  -  log  j*) 

as  the  expression  for  the  difference  of  levels,  in  centimetres.  It  is 
usual  to  put  for  t  the  arithmetical  mean  of  the  temperatures  at  the 
two  stations. 

The  determination  of  heights  by  means  of  atmospheric  pressure, 
whether  the  pressure  be  observed  directly  by  the  barometer,  or  in- 
directly by  the  boiling-point  thermometer  (which  will  be  described 
in  Part  II.),  is  called  hypsometry  (v^oc,  height). 

As  a  rough  rule,  it  may  be  stated  that,  in  ordinary  circumstances, 
the  barometer  falls  an  inch  in  ascending  900  feet. 

214.  Diurnal  Oscillation  of  the  Barometer. — In  these  latitudes,  the 
mercurial  column  is  in  a  continual  state  of  irregular  oscillation;  but 
in  the  tropics  it  rises  and  falls  with  great  regularity  according  to  the 
hour  of  the  day,  attaining  two  maxima  in  the  twenty-four  hours. 

It  generally  rises  from  4  A.M.  to  10  A.M.,  when  it  attains  its  first 
maximum;  it  then  falls  till  4  P.M.,  when  it  attains  its  first  minimum; 
a  second  maximum  is  observed  at  10  P.M.,  and  a  second  minimum  at 
4  A.M.  The  hours  of  maxima  and  minima  are  called  the  tropical 
hours  (TTMTTW,  to  turn),  and  vary  a  little  with  the  season  of  the  year. 
The  difference  between  the  highest  maximum  and  lowest  minimum 
is  called  the  diurnal1  range,  and  the  half  of  this  is  called  the  ampli- 

1  The  epithets  annual  and  diurnal,  when  prefixed  to  the  words  rariation,  range,  ampli- 
tude, denote  the  period  of  the  variation  in  question ;  that  is,  the  time  of  a  complete  oscilla- 
tion.    Diurnal  variation  does  not  denote  variation  from  one  day  to  another,  but  the  varia- 
tion which  goes  through  its  cycle  of  values  in  one  day  of  twenty-four  hours.     Annual 
11 


162 


VARIATIONS   OF   THE   BAROMETER. 


tude  of  the  diurnal  oscillation.     The  amount  of  the  former  does  not 
exceed  about  a  tenth  of  an  inch. 

The  character  of  this  diurnal  oscillation  is  represented  in  Fig.  119. 
The  vertical  lines  correspond  to  the  hours  of  the  day;  lengths  have 
been  measured  upwards  upon  them  proportional  to  the  barometric 
heights  at  the  respective  hours,  diminished  by  a  constant  quantity; 
and  the  points  thus  determined  have  been  connected  by  a  continuous 
curve.  It  will  be  observed  that  the  two  lower  curves,  one  of  which 
relates  to  Cumana,  a  town  of  Venezuela,  situated  in  about  10°  north 
latitude,  show  strongly  marked  oscillations 
corresponding  to  the  maxima  and  minima.  In 
our  own  country,  the  regular  diurnal  oscilla- 
tion is  masked  by  irregular  fluctuations,  so 
that  a  single  day's  observations  give  no  clue 
to  its  existence.  Nevertheless,  on  taking 
observations  at  regular  hours  for  a  number  of 
consecutive  days,  and  comparing  the  mean 


Fig.  119. 
Curves  of  i>iurual  Variation. 


2  IB  is  21  24  heights  for  the  different  hours,  some  indications 
of  the  law  will  be  found.  A  month's  observa- 
tions will  be  sufficient  for  an  approximate 
indication  of  the  law;  but  observations  extending  over  some  years 
will  be  required,  to  establish  with  anything  like  precision  the  hours 
of  maxima  and  the  amplitude  of  the  oscillation. 

The  two  upper  curves  represent  the  diurnal  variation  of  the  baro- 
meter at  Padua  (lat.  45°  24')  and  Abo  (lat.  60°  56'),  the  data  having 
been  extracted  from  Kaemtz's  Meteorology.  We  see,  by  inspection 
of  the  figure,  that  the  oscillation  in  question  becomes  less  strongly 
marked  as  the  latitude  increases.  The  range  at  Abo  is  less  than 
half  a  millimetre.  At  about  the  70th  degree  of  north  latitude  it 
becomes  insensible;  and  in  approaching  still  nearer  to  the  pole,  it 
appears  from  observations,  which  however  need  further  confirmation, 
that  the  oscillation  is  reversed;  that  is  to  say,  that  the  maxima  here 
are  contemporaneous  with  the  minima  in  lower  latitudes. 

There  can  be  little  doubt  that  the  diurnal  oscillation  of  the 
barometer  is  in  some  way  attributable  to  the  heat  received  from  the 
sun,  which  produces  expansion  of  the  air,  both  directly,  as  a  mere 

range  denotes  the  range  that  occurs  within  a  year.     This  rule  is  universally  observed  by 
writers  of  high  scientific  authority. 

A  table,  exhibiting  the  values  of  an  element  for  each  month  in  the  year,  is  a  table  of 
annual  (not  monthly)  variation ;  or  it  may  be  more  particularly  described  as  a  table  of 
variations  from  month  to  month. 


PREDICTION    OF   WEATHER.  1C3 

consequence  of  heating,  and  indirectly,  by  promoting  evaporation; 
but  the  precise  nature  of  the  connection  between  this  cause  and  the 
diurnal  barometric  oscillation  has  not  as  yet  been  satisfactorily 
established. 

215.  Irregular  Variations  of  the  Barometer. — The  height  of  the 
barometer,  at  least  in  the  temperate  zones,  depends  on  the  state  of 
the  atmosphere;  and  its  variations  often  serve  to  predict  the  changes 
of  weather  with  more  or  less  certainty.     In  this  country  the  baro- 
meter generally  falls  for  rain  or  S.W.  wind,  and  rises  for  fine  weather 
or  N.E.  wind. 

Barometers  for  popular  use  have  generally  the  words — 

Set  fair.  Fair.  Change.  Rain.        Much  rain.         Stormy. 

marked  at  the  respective  heights 

30-5  30  29-5  29  28'5  28  inches. 

These  words  must  not,  however,  be  understood  as  absolute  predic- 
tions. A  low  barometer  rising  is  generally  a  sign  of  fine,  and  a  high 
barometer  falling  of  wet  weather.  Moreover,  it  is  to  be  borne  in 
mind  that  the  barometer  stands  about  a  tenth  of  an  inch  lower  for 
every  hundred  feet  that  we  ascend  above  sea-level. 

The  connection  between  a  low  or  falling  barometer  and  wet 
weather  is  to  be  found  in  the  fact  that  moist  air  is  specifically 
lighter  than  dry,  even  at  the  same  temperature,  and  still  more  when, 
as  usually  happens,  moist  air  is  warmer  than  dry. 

Change  of  wind  usually  begins  in  the  upper  regions  of  the  air 
and  gradually  extends  downwards  to  the  ground;  hence  the  baro- 
meter, being  affected  by  the  weight  of  the  whole  superincumbent 
atmosphere,  gives  early  warning. 

216.  Weather   Charts.     Isobaric   Lines.— The   extension    of    tele- 
graphic communication  over  Europe  has  led  to  the  establishment  of 
a  system  of  correspondence,  by  which  the  barometric  pressures,  at  a 
given  moment,  at  a  number  of  stations  which  have  been  selected  for 
meteorological  observation,  are   known   at   one   or   more  stations 
appointed  for  receiving  the  reports.     From  the  information  thus 
furnished,  curves  (called  isobaric  lines,  or  isobars)  are  dfawn,  upon 
a  chart,  through  those  places  at  which  the  pressure  is  the  same. 
The  barometric  condition  of  an  extensive  region  is  thus  rendered 
intelligible  at  a  glance.     Plate   I.  is  a  specimen  of  one  of  these 


IQ4,  VARIATIONS   OF   THE   BAROMETER. 

charts,1  prepared  at  the  observatory  of  Paris;  it  refers  to  the  22d  of 
January,  1868.  Besides  the  isobaric  lines,  the  charts  indicate,  by 
the  system  of  notation  explained  at  the  left  of  the  figure,  the  general 
state  of  the  weather,  the  strength  of  wind,  and  state  of  the  sea.  The 
isobaric  curves  correspond  to  differences  of  five  millimetres  (about 
0-2  inch)  of  pressure,  and  according  as  they  are  near  together  or  far 
apart  the  variation  of  pressure  in  passing  from  one  to  another  is 
more  or  less  sudden  (or  to  use  a  very  expressive  modern  phrase,  the 
barometric  gradient  is  more  or  less  steep),  just  as  the  contour  lines 
on  a  map  of  hillv  ground  approach  each  other  most  nearly  where 
the  ground  is  steepest.  Charts  on  the  same  general  plan  are  issued 
daily  from  the  Meteorological  Office  in  London. 

A  steep  barometric  gradient  tends  to  produce  a  strong  wind.  It 
will  be  observed,  however,  from  the  arrows  on  the  chart,  that  the 
direction  of  the  wind,  instead  of  being  coincident  with  the  line  of 
steepest  descent  from  each  isobar  to  the  next  below  it,  generally 
makes  a  large  angle,  considerably  exceeding  45°,  to  the  right  of  it. 
In  the  southern  hemisphere  the  deviation  is  to  the  left  instead  of  to 
the  right.  This  law,  known  as  Buys  Ballot's,  is  found  to  hold  in 
almost  every  instance,  and  is  dependent  on  the  earth's  rotation.2 

The  isobars  frequently,  as  in  the  example  here  selected,  form 
closed  curves  encircling  a  region  of  barometric  depression.  Two 
such  centres  are  here  exhibited — one  in  the  south  of  England  and 
the  other  in  the  west  of  Russia.  Great  atmospheric  disturbances 
are  always  accompanied  by  such  centres  of  depression.  The  air,  in 
fact,  rushes  in  from  all  sides,  usually  with  a  spiral  motion,  towards 
these  centres,  the  direction  of  rotation  in  the  spiral  being,  for  the 
northern  hemisphere,  opposite  to  the  motion  of  the  hands  of  a  watch 
lying  with  its  face  upwards.  The  centrifugal  force  due  to  this 
rotation  tends  to  increase  the  original  central  depression,  and  thus 
protracts  the  duration  of  the  phenomenon. 

1  The  curves  drawn  upon  this  chart  are  isobaric  lines,  each  corresponding  to  a  particular 
barometric  pressure,  which  is  indicated  by  the  numerals  marked  against  it.     These  denote 
the  pressure  in  millimetres  diminished  by  700.     For  example,  the  line  which  passes 
through  the  south  of  Spain  corresponds  to  the  pressure  770  millimetres;  that  through  the 
north  of  Spain  to  765  millimetres.     The  curves  are  drawn  for  every  fifth  millimetre. 
The  smaller  numerals,  which  are  given  to  one  place  of  decimals,  indicate  the  pressures 
actually  observed  at  the  different  stations,  from  which  the  isobaric  lines  are  drawn  by 
estimation. 

The  other  symbols  refer  to  cloud,  wind,  and  sea,  and  are  explained  at  the  left  of  the  chart. 

2  The  influence  of  the  earth's  rotation  in  modifying  the  direction  of  winds  is  discussed 
in  a  paper  "  On  the  General  Circulation  and  Distribution  of  the  Atmosphere,"  by  the 
editor  of  this  work,  in  the  Philosophical  Magazine  for  September,  1871. 


ISOBARS   AND   WEATHER   CHARTS.  165 

These  revolving  storms  are  called  cyclones.  They  attain  their 
greatest  violence  in  tropical  regions,  the  West  Indies  being  especially 
noted  for  their  destructive  effect.  They  frequently  proceed  from  the 
Gulf  of  Mexico  in  a  north-easterly  direction,  increasing  in  diameter 
as  they  proceed,  but  diminishing  in  violence.  Their  velocity  of 
translation  is  usually  from  ten  to  twenty  miles  an  hour. 

Storm-warnings  are  based  partly  upon  information  received  by 
telegraph  of  storms  that  have  actually  commenced,  and  partly  upon 
barometric  gradients.1 

1  For  fuller  information  respecting  the  laws  of  storms,  which  is  a  purely  modern  subject, 
and  is  continually  receiving  fresh  developments,  we  would  refer  to  Mr.  Buchan's  llandy 
Book  of  Meteorology.  See  also  the  last  chapter  of  Part  II.  of  the  present  Work. 


CHAPTER  XIX. 

BOYLE'S  (OR  MARIOTTE'S)  LAW.1 


217.  Boyle's  Law. — All  gases  exhibit  a  continual  tendency  to  ex- 
pand, and  thus  exert  pressure  against  the  vessels  in  which  they  are 
confined.     The  intensity  of  this  pressure  depends  upon  the  volume 
which  they  occupy,  increasing  as   this  volume  diminishes.     By  a 
number  of  careful  experiments  upon  this  point,  Boyle  and  Mariotte 
independently  established  the  law  that  this  pressure  varies  inversely 
as  the  volume,  provided  that  the  temperature  remain  constant.     As 
the  density  also  varies  inversely  as  the  volume,  we  may  express  the 
law  in  other  words  by  saying  that  at  the  same  temperature  the 
density  varies  directly  as  the  pressure. 

If  V  and  V  be  the  volumes  of  the  same  quantity  of  gas,  P  and  F, 
D  and  D',  the  corresponding  pressures  and  densities,  Boyle's  law  will 
be  expressed  by  either  of  the  equations 

P      V       P       D 
F~V'     F  =  D'' 

218.  Boyle's  Tube. — The  correctness  of  this  law  may  be  verified 
by  means  of  the  following  apparatus,  which  was  employed  by  both 
the  experimenters  above  named.     It  consists  (Fig.  120)  of  a  bent 
tube  with  branches  of  unequal  length;   the  long  branch  is  open, 
and  the  short  branch  closed.      The  tube  is  fastened  to  a  board 
provided  with  two  scales,  one  by  the  side  of  each  branch.     The 

1  Boyle,  in  his  Defence  of  the  Doctrine  touching  the  Spring  and  Weight  of  the  Air  against 
the  Objections  of  Francisc.us  Linus,  appended  to  New  Experiments,  Physico-mechanical,  &c. 
(second  edition,  4to,  Oxford,  1662),  describes  the  two  kinds  of  apparatus  represented  in 
Figs.  120,  121  as  having  been  employed  by  him,  and  gives  in  tabular  form  the  lengths  of 
tube  occupied  by  a  body  of  air  at  various  pressures.  These  observed  lengths  he  compares 
with  the  theoretical  lengths  computed  on  the  assumption  that  volume  varies  reciprocally 
as  pressure,  and  points  out  that  they  agree  within  the  limits  of  experimental  error. 

Manotte's  treatise,  De  la  Nature  de  I' Air,  is  stated  in  th«  Bioyraphie  UnuerseUe  to  have 
been  published  in  1679.  (See  Preface  to  Tait's  Thermodynamics,  p.  iv.) 


BOYLE'S  TUBE. 


167 


graduation  of  both  scales  begins  from  the  same  horizontal  line 
through  0,  0.  Mercury  is  first  poured  in  at  the  extremity  of  the 
long  branch,  and  by  inclining  the  apparatus  to  either  side,  and 
cautiously  adding  more  of  the  liquid  if  required,  the  mercury 
can  be  made  to  stand  at  the  same  level  in  both 
branches,  and  at  the  zero  of  both  scales.  Thus 
we  have,  in  the  short  branch,  a  quantity  of  air 
separated  from  the  external  air,  and  at  the  same 
pressure.  Mercury  is  then  poured  into  the  long 
branch,  so  as  to  reduce  the  volume  of  this  inclosed 
air  by  one-half;  it  will  then  be  found  that  the 
difference  of  level  of  the  mercury  in  the  two 
branches  is  equal  to  the  height  of  the  barometer 
at  the  time  of  the  experiment;  the  compressed  air 
therefore  exerts  a  pressure  equal  to  that  of  two 
atmospheres.  If  more  mercury  be  poured  in  so  as 
to  reduce  the  volume  of  the  air  to  one-third  or  one- 
fourth  of  the  original  volume,  it  will  be  found  that 
the  difference  of  level  is  respectively  two  or  three 
times  the  height  of  the  barometer;  that  is,  that  the 
compressed  air  exerts  a  pressure  equal  respectively 
to  that  of  three  or  four  atmospheres.  This  ex- 
periment therefore  shows  that  if  the  volume  of  the 
gas  becomes  two,  three,  or  four  times  as  small,  the 
pressure  becomes  two,  three,  or  four  times  as  great. 
This  is  the  principle  expressed  in  Boyle's  law. 

The  law  may  also  be  verified  in  the  case  where 
the  gas  expands,  and  where  its  pressure  conse- 
quently diminishes.  For  this  purpose  a  barometric 
tube  (Fig.  121),  partially  filled  with  mercury,  is* 
inverted  in  a  tall  vessel,  containing  mercury  also, 
and  is  held  in  such  a  position  that  the  level  of  the 
liquid  is  the  same  in  the  tube  and  in  the  vessel. 
The  volume  occupied  by  the  gas  is  marked,  and  the  tube  is  raised; 
the  gas  expands,  its  pressure  diminishes,  and,  in  virtue  of  the  excess 
of  the  atmospheric  pressure,  a  column  of  mercury  ab  rises  in  the 
tube,  such  that  its  height,  added  to  the  pressure  of  the  expanded  air, 
is  equal  to  the  atmospheric  pressure.  It  will  then  be  seen  that  if 
the  volume  of  air  becomes  double  wThat  it  was  before,  the  height  of 
the  column  raised  is  one-half  that  of  the  barometer;  that  is,  the 


1C8  BOYLE'S  (OR  MARIOTTE'S)  LAW. 

expanded  air  exerts  a  pressure  equal  to  half  that  of  the  atmosphere. 
If  the  volume  is  trebled,  the  height  of  the  column  is  two-thirds  that 
of  the  barometer;  that  is,  the  pressure  of  the 
expanded  air  is  one-third  that  of  the  atmosphere, 
a  result  in  accordance  with  Boyle's  law. 

219.  Despretz's  Experiments. — The  simplicity  of 
Boyle's  law,  taken  in  conjunction  with  its  apparent 
agreement  with  facts,  led  to  its  general  acceptance 
as  a  rigorous  truth  of  nature,  until  in  1825 
Despretz  published  an  account  of  experiments, 
showing  that  different  gases  are  unequally  com- 
pressible. He  inverted  in  a  cistern  of  mercury 
several  cylindrical  tubes  of  equal  height,  and 
filled  them  with  different  gases.  The  whole 
apparatus  was  then  inclosed  in  a  strong  glass 
vessel  filled  with  water,  and  having  a  screw  piston 
as  in  CErsted's  piesometer  (§  130).  On  pressure 
being  applied,  the  mercury  rose  to  unequal  heights 
in  the  different  tubes,  carbonic  acid  for  example 
being  more  reduced  in  volume  than  air.  These 
experiments  proved  that  even  supposing  Boyle's 
law  to  be  true  for  one  of  the  gases  employed, 
U  could  not  be  rigorously  true  for  more  than 
one. 

In  1829  Dulong  and  Arago  undertook  a  laborious  series  of  experi- 
ments with  the  view  of  testing  the  accuracy  of  the  law  as  applied  to 
air;  and  the  results  which  they  obtained,  even  when  the  pressure  was 
increased  to  twenty-seven  atmospheres,  agreed  so  nearly  with  it  as 
to  confirm  them  in  the  conviction  that,  for  air  at  least,  it  was  rigor- 
ously true.  When  re-examined,  in  the  light  of  later  researches,  the 
results  obtained  by  Dulong  and  Arago  seem  to  point  to  a  different 
conclusion. 

220.  Unequal  Compressibility  of  Different  Gases.— The  unequal 
compressibility  of  different  gases,  which  was  first  established  by 
Despretz's  experiments  above  described,  is  now  usually  exhibited  by 
the  aid  of  an  apparatus  designed  by  Pouillet  (Fig.  122).  A  is  a  cast- 
iron  reservoir,  containing  mercury  surn^unted  by  oil.  In  this  latter 
liquid  dips  a  bronze  plunger  P,  the  upper  part  of  which  has  a  thread 
cut  upon  it,  and  works  in  a  nut,  so  that  the  plunger  can  be  screwed 
up  or  down  by  means  of  the  lever  L.  The  reservoir  A  communicates 


NOT   RIGOROUSLY  EXACT. 


169 


by  an  iron  tube  with  another  cast-iron  vessel,  into  which  are  firmly 
fastened  two  tubes  TT  about  six  feet  in  length  and  -j^th  of  an  inch 
in  internal  diameter,  very  carefully  calibrated.  Equal 
volumes  of  two  gases,  perfectly  dry,  are  introduced  into 
these  tubes  through  their  upper  ends,  which  are  then 
hermetically  sealed.  The  plunger  is  then  made  to 
descend,  and  a  gradually  increasing  pressure  is  exerted, 
the  volumes  occupied  by  the  gases  are  measured,  and  it 
is  ascertained  that  no  two  gases  follow  precisely  the 
same  law  of  compression.  The  difference,  however,  is 
almost  insensible  when  the  gases  employed  are  those 
which  are  very  difficult  to  liquefy,  as  air,  oxygen, 
hydrogen,  nitrogen,  nitric  oxide,  and  marsh -gas.  But 
when  we  compare  any  one  of  these  with  one  of  the 
more  liquefiable  gases,  such  as  carbonic  acid,  cyanogen, 
or  ammonia,  the  difference  is  rapidly  and  distinctly 
manifested.  Thus,  under  a  pressure  of  twenty-five 
atmospheres,  carbonic  acid  occupies  a  volume  which  is 
only  £ ths  of  that  occupied  by  air. 

221.  Regnault's  Experiments. — Boyle's  law,  therefore, 
is  not  to  be  considered  as  rigorously  exact;  but  it  is  so 
nearly  exact  that  to  demon- 
strate its  inaccuracy  for  one 
of  the  more  permanent  gases, 
and  still  more  to  determine 
the  law  of  deviation  for  each 
gas,  very  precise  methods  of 
measurement  are  necessary. 
Jn  ordinary  experiments  on 
compression,  and  even  in  the 
elaborate  investigations  of 
Dulong  and  Arago,  a  definite 
portion  of  gas  is  taken  and 
successively  diminished  in 
volume  by  the  application  of 
continually  increasing  pres- 
sure. In  experiments  of 
this  kind,  as  the  pressure 
increases,  the  volume  under  measurement  becomes  smaller,  and  the 
precision  with  which  it  can  be  measured  consequently  diminishes. 


Fig.  122.-Po 


.ibUity  of  Ditfe 


ing  Unequal 


170 


BOYLE'S  (OB  MARIOTTE'S)  LAW. 


Regnault  adopted  the  plan  of  operating  in  all  cases  upon  the  same 
volume  of  gas,  which  being  initially  at  different  pressures, 
was  always  reduced  to  one-half.  The  pressure  was 
observed  before  and  after  this  operation,  and,  if  Boyle's 
law  were  true,  its  value  should  be  found  to  be  doubled. 
In  this  way  the  same  precision  of  measurement  is  obtained 
at  high  as  at  low  pressures. 

A  general  view  of  Regnault's  apparatus  is  given  in 
Fig.  123.     There  is  an  iron  reservoir  containing  mercury, 
furnished  at  the  top  with  a  force-pump  for  water.     The 
lower  part  of  this  reservoir  communicates  with  a  cylinder 
which  is  also  of  iron,  and  in  which  are  two  openings  to 
admit  tubes.    Communication  between  the  reservoir  and 
the  cylinder  can  be  established  or  interrupted  by  means 
of  a  stop-cock  R,  of  very  exact  workmanship.     Into  one 
of  the  openings  is  fitted  the 
lowest  of  a  series  of  glass  tubes 
A,  which  are  placed  end  to  end, 
and  firmly  joined  to  each  other 
by  metal    fittings,   so    as    to 
form    a    vertical    column    of 
about    twenty-five   metres    in 
height. 

The  height  of  the  mercurial 
column  in  this  long  mano- 
metric  tube  could  be  exactly 


Fig.  1C3.— Regnault's  Apparatus  for  Testing  Boyle's  Law. 


REGNAULT'S   INVESTIGATIONS.  171 

determined  by  means  of  reference  marks  placed  at  distances  of 
about  '95  of  a  metre,  and  by  the  graduation  on  the  tubes  forming 
the  upper  part  of  the  column.  The  mean  temperature  of  the 
mercurial  column  was  given  by  thermometers  placed  at  different 
heights.  Into  the  second  opening  in  the  cylinder  fits  the  lower 
extremity  of  the  tube  B,  which  is  divided  into  millimetres,  and  also 
gauged  with  great  accuracy.  This  tube  has  at  its  upper  end  a  stop- 
cock r  which  can  open  communication  with  the  reservoir  V,  into 
which  the  gas  to  be  operated  on  is  forced  and  compressed  by  means 
of  the  pump  P. 

An  outer  tube,  which  is  not  shown  in  the  figure,  envelops  the 
tube  B,  and,  being  kept  full  of  water,  which  is  continually  renewed, 
enables  the  operator  to  maintain  the  tube  at  a  temperature  sensibly 
constant,  which  is  indicated  by  a  very  delicate  thermometer.  Before 
fixing  the  tube  in  its  place,  the  point  corresponding  to  the  middle  of 
its  volume  is  carefully  ascertained,  and  after  the  tube  has  been  per- 
manently fixed,  the  distance  of  this  point  from  the  nearest  of  the 
reference  marks  is  observed.1 

After  these  explanatory  remarks  we  may  describe  the  mode  of 
conducting  the  experiments.  The  gas  to  be  operated  on,  after  being 
first  thoroughly  dried,  was  introduced  at  the  upper  part  of  the  tube  B, 
the  stop-cock  of  the  pump  being  kept  open,  so  as  to  enable  the  gas 
to  expel  the  mercury  and  occupy  the  entire  length  of  the  tube.  The 
force-pump  was  then  brought  into  play,  and  the  gas  was  reduced  to 
about  half  of  its  former  volume;  the  pressure  in  both  cases  being 
ascertained  by  observing  the  height  of  the  mercury  in  the  long  tube 
above  the  nearest  mark.  It  is  important  to  remark  that  it  is  not  at 
all  necessary  to  operate  always  upon  exactly  the  same  initial  volume, 
and  reduce  it  exactly  to  one-half,  which  would  be  a  very  tedious 
operation;  these  two  conditions  are  approximately  fulfilled,  and  the 
graduation  of  the  tube  enables  the  observer  always  to  ascertain  the 
actual  volumes. 
>  222.  Results. — The  general  result  of  the  investigations  of  Regnault 

1  Regnault's  apparatus  was  fixed  in  a  small  square  tower  of  about  fifteen  metres  in 
height,  forming  part  of  the  buildings  of  the  College  de  France,  and  which  had  formerly 
been  built  by  Savart  for  experiments  in  hydraulics.  The  tower  could  therefore  contain 
only  the  lower  part  of  the  manometric  column ;  the  upper  part  rose  above  the  platform  at 
the  top  of  the  tower,  resting  against  a  sort  of  mast  which  could  be  ascended  by  the  ob- 
server. The  readings  inside  the  tower  could  be  made  by  means  of  a  cathetometer,  but  this 
was  impossible  in  the  upper  portion  of  the  column,  and  for  this  reason  the  tubes  forming 
this  portion  were  graduated. — D. 


172  BOYLE'S  (OR  MARIOTTE'S)  LAW. 

is,  that  Boyle's  law  does  not  exactly  represent  the  compressibility 
even  of  air,  hydrogen,  or  nitrogen,  which,  with  carbonic  acid,  were 
the  gases  operated  on  by  him.  He  found  that  for  all  the  gases  on 
which  he  operated,  except  hydrogen,  the  product  VP  of  the  volume 
and  pressure,  instead  of  remaining  constant,  as  it  would  if  Boyle's 
law  were  exact,  diminished  as  the  compression  was  increased.  This 
diminution  is  particularly  rapid  in  the  cases  of  the  more  liquefiable 
gases,  such  as  carbonic  acid,  at  least  when  the  experiments  are  con- 
ducted at  ordinary  atmospheric  temperatures.  The  lower  the  tem- 
perature, the  greater  is  the  departure  from  Boyle's  law  in  the  case 
of  these  gases.  For  hydrogen,  he  found  the  departure  from  Boyle's 
law  to  be  in  the  opposite  direction; — the  product  VP  increased  as 
the  gas  was  more  compressed. 

223.  Manometers  or  Pressure-gauges.— Manometers  or  pressure- 
gauges  are  instruments  for  measuring  the  elastic  force  of  a  gas  or 
vapour  contained  in  the  interior  of  a  closed  space.  This  elastic  force 
is  generally  expressed  in  units  called  atmospheres  (§  198),  and  is  often 
measured  by  means  of  a  column  of  mercury. 

When  one  end  of  the  column  of  mercury  is  open  to  the  air,  as  in 
Regnault's  experiments  above  described,  the  gauge  is  called  an  open 
mercurial  gauge. 

The  open  mercurial  pressure-gauge  is  often  used  in  the  arts  to 
measure  pressures  which  are  not  very  considerable.  Fig.  124  repre- 
sents one  of  its  simplest  forms.  The  apparatus  consists  of  a  box, 
generally  of  iron,  at  the  top  of  which  is  an  opening  closed  by  a  screw 
stopper,  which  is  traversed  by  the  tube  6,  open  at  both  ends,  and 
dipping  into  the  mercury  in  the  box.  The  air  or  vapour  whose 
elastic  force  is  to  be  measured  enters  by  the  tube  a,  and  presses  upon 
the  mercury.  It  is  evident  that  if  the  level  of  the  liquid  in  the  box 
is  the  same  as  in  the  tube,  the  pressure  in  the  box  must  be  exactly 
equal  to  that  of  the  atmosphere.  If  the  mercury  in  the  tube  rises 
above  that  in  the  box,  the  pressure  of  the  air  in  the  box  must  exceed 
that  of  the  atmosphere  by  a  pressure  corresponding  to  the  height  of 
the  column  raised.  The  pressures  are  generally  marked  in  atmo- 
spheres upon  a  scale  beside  the  tube. 

-,  224.  Multiple  Branch  Manometer.— When  the  pressures  to  be  mea- 
sured are  considerable,  as  in  the  boiler  of  a  high-pressure  steam- 
engine,  the  above  instrument,  if  employed  at  all,  must  be  of  a  length 
corresponding  to  the  pressure.  If,  for  instance,  the  pressure  in  ques- 
tion is  eight  atmospheres,  the  length  of  the  tube  must  be  at  least 


MANOMETERS. 


173 


8  X  30  inches= 20  feet.  Such  an  arrangement  is  inconvenient  even  for 
stationary  machines,  and  is  entirely  inapplicable  to  movable  machines. 

Without  departing  from  the  principle  of  the  open  mercurial  pres- 
sure-gauge, namely,  the  balancing  of  the  pressure  to  be  observed 
against  the  weight  of  a  liquid  increased  by  one  atmosphere,  we  may 
reduce  the  length  of  the  instrument  by  an  artifice  already  employed 
by  Fahrenheit  in  his  barometer  (§  207). 

The  apparatus  for  this  purpose  consists  of  an  iron  tube  ABCD 


Fig.  124.— Open 
Mercurial  Manometer. 


Fig  125.  — Multiple  Branch  Manometer. 


(Fig.  125)  bent  back  upon  itself  several  times.  The  extremity  A 
communicates  with  the  boiler  by  a  stop-cock,  and  the  last  branch 
CD  is  of  glass,  with  a  scale  by  its  side. 

The  first  step  is  to  fill  the  tube  with  mercury  as  far  as  the  level 
MN.  At  this  height  are  holes  by  which  the  mercury  escapes  when 
it  reaches  them,  and  which  are  afterwards  hermetically  sealed.  The 
upper  portions  are  filled  with  water  through  openings  which  are  also 
stopped  after  the  tube  has  been  filled.  If  the  mercury  in  the  first 
tube,  which  is  in  communication  with  the  reservoir  of  gas,  falls 
through  a  distance  h,  it  will  alternately  rise  and  fall  through  the  same 
distance  in  the  other  tubes.  The  difference  of  pressure  between 
the  two  ends  of  the  gauge  is  represented  by  the  weight  of  a  column 
of  mercury  of  height  10k  diminished  by  the  weight  of  a  column  of 
water  of  height  8h.  Reduced  to  mercury,  the  difference  of  pressure 

is  therefore  10A  -         =  Q'4h. 


174  BOYLE'S  (OR  MARIOTTE'S)  LAW. 

225.  Compressed-air  Manometer.  —  This  instrument,  which  may  as- 
sume different  forms,  sometimes  consists,  as  in  Fig.  126,  of  a  bent 
tube  AB  closed  at  one  end  a,  and  containing  within  the  space  Act  a 
quantity  of  air,  which  is  cut  off  from  external  communication  by  a 
column  of  mercury.  The  apparatus  has  been  so  constructed,  that 
when  the  pressure  on  B  is  equal  to  that  of  the  atmosphere,  the  mer- 
cury stands  at  the  same  height  in  both  branches;  so  that,  under 
these  circumstances,  the  inclosed  air  is  exactly  at  atmospheric  pres- 
sure. But  if  the  pressure  increases,  the  mercury  is  forced  into  the 
left  branch,  so  that  th'e  air  in  that  branch  is  compressed,  until  equi- 
librium is  established.  The  pressure  exerted  by 
the  gas  at  B  is  then  equal  to  the  pressure  of  the 
compressed  air,  together  with  that  of  a  column  of 
mercury  equal  to  the  difference  of  level  of  the  liquid 
in  the  two  branches.  This  pressure  is  usually 
expressed  in  atmospheres  on  the  scale  ab. 

The  graduation  of  this  scale  is  effected  empiri- 
cally in  practice,  by  placing  the  manometer  in 
communication  with  a  reservoir  of  compressed  air 
wnose  pressure  is  given  by  an  open  mercurial  gauge, 
or  by  a  standard  manometer  of  any  kind. 

If  the  tube  AB  be  supposed  cylindrical,  the  graduation  can  be 
calculated  by  an  application  of  Boyle's  law. 

Let  I  be  the  length  of  the  tube  occupied  by  the  inclosed  air  when 
its  pressure  is  equal  to  that  of  one  atmosphere;  at  the  point  to  which 
the  level  of  the  mercury  rises  is  marked  the  number  1.  It  is  required 
to  find  what  point  the  end  of  the  liquid  column  should  reach  when 
a  pressure  of  n  atmospheres  is  exerted  at  B.  Let  x  be  the  height  of 
this  point  above  1;  then  the  volume  of  the  air,  which  was  originally  I, 
has  become  I  -  x,  and  its  pressure  is  therefore  equal  to  H  -—^  H  being 
the  mean  height  of  the  barometer.  This  pressure,  together  with  that 
due  to  the  difference  of  level  2x,  is  equivalent  to  n  atmospheres. 
We  have  thus  the  equation  — 

Hr^  +  2*  =  n11' 
whence 

2**-(nH  +  20*+  (n-l)Hi  =  0. 
r  _  nH  +  <U  ±  y  (MH  +  2/)«  -  8  (»  -  1)  ffi 


We  thus  find  two  values  of  0;  but  that  given  by  taking  the  positive 


MANOMETERS.  175 

sign  of  the  radical  is  inadmissible;  for  if  we  put  7i=l,  we  ought  to 
have  x=o,  which  will  not  be  the  case  unless  the  sign  of  the  radical 
is  negative. 

By  giving  n  the  successive  values  1|,  2,  2|,  3,  &c.,  in  this 
expression  for  x,  we  find  the  points  on  the  scale  corresponding  to 
pressures  of  one  atmosphere  and  a  half,  two  atmospheres,  &c. 

As  the  pressure  increases,  the  distance  traversed  by  the  mercury 
for  an  increment  of  pressure  equal  to  one  atmosphere  becomes 
continually  less,  and  the  sensibility  of  the  instrument  accordingly 
decreases.  This  inconvenience  is  partly  avoided  by  the  arrange- 
ment shown  in  Fig.  127.  The  branch  containing  the  air  is  made 
tapering  so  that,  as  the  mercury  rises,  equal  changes  of  volume 
correspond  to  increasing  lengths. 

226.  Metallic  Manometers. — The  fragility  of  glass  tubes,  and  the 
fact  that  they  are  liable  to  become  opaque  by  the  mercury  clinging 


I 

Fig.  127.— Compressed  air 

Manometer.  Fig.  128. — Bourdon's  Pressure  gauge. 

to  their  sides,  are  serious  drawbacks  to  their  use,  especially  in 
machines  in  motion.  Accordingly,  metallic  manometers  are  often 
employed,  their  indications  depending  upon  changes  of  form  effected 
by  the  pressure  of  gas  on  its  containing  vessel.  We  shall  here  men- 
tion only  Bourdon's  gauge  (Fig.  128).  It  consists  essentially  of  a 
copper  tube  of  elliptic  section,  which  is  bent  through  about  540°,  as 
represented  in  the  figure.  One  of  the  extremities  communicates  by 
a  stop-cock  with  the  reservoir  of  steam  or  compressed  gas;  to  the 
other  extremity  is  attached  a  steel  needle  which  traverses  a  scale. 
When  the  pressure  is  the  same  within  and  without  the  tube  the  end 
of  the  needle  stands  at  the  mark  1 ;  but  if  the  pressure  within  the 


176 


BOYLE'S  (OR  MARIOTTE'S)  LAW. 


tube  increases,  the  curvature  diminishes,  the  free  extremity  of  the 
tube  moves  away  from  the  fixed  extremity,  and  the  needle  traverses 
the  scale. 

227.  Mixture  of  Gases.— When  gases  of  different  densities  are 
inclosed  in  the  same  space,  experiment  shows  that,  even  under  the 
most  unfavourable  circumstances,  an 
intimate  mixture  takes  place,  so  that 
each  gas  becomes  uniformly  diffused 
through  the  entire  space.  This  fact 
has  been  shown  by  a  decisive  ex- 
periment due  to  Berthollet.  He  took 
two  globes  (Fig.  129)  which  could 
be  screwed  together,  and  placed  them 
in  a  cellar.  The  lower  globe  was 
filled  with  carbonic  acid,  the  upper 
globe  with  hydrogen.  Communication 
was  established  between  them,  and 
after  some  time  it  was  ascertained 
that  the  gases  had  become  uniformly 
mixed;  the  proportions  being  the 
same  in  both  globes.  Gaseous  diffu- 
sion is  a  comparatively  rapid  process. 
The  diffusion  of  liquids,  when  not  assisted  by  gravity,  is,  on  the 
other  hand,  exceedingly  slow. 

If  several  gases  are  inclosed  in  the  same  space,  each  of  them 
exerts  the  same  pressure  as  if  the  others  were  absent,  in  other 
words,  the  pressure  exerted  by  the  mixture  is  equal  to  the  sum  of 
the  pressures  which  each  would  exert  separately.  This  is  known 
as  "Dalton's  law  for  gaseous  mixtures."  The  separate  pressures 
can  easily  be  calculated  by  Boyle's  law,  when  the  original  pressure 
and  volume  of  each  gas  are  known. 

For  example,  let  V  and  P,  V  and  F,  V"  and  P"  be  the  volumes 
and  pressures  of  the  gases  which  are  made  to  pass  into  a  vessel  of 
volume  U.  The  first  gas  exerts,  when  in  this  vessel,  a  pressure 


Fig.  129.— Mixture  of  Gases. 


VP 


equal  to  -^r,  the  second  a  pressure  equal  to  -^-,  the  third  a  pressure 
equal  to    -^.-    and  so  on,  so  that  the  total  pressure  M  is  equal  to 

VP      VP'      V"P" 

U  +  ~tf  +  ~\T'  wnence  MU  =  VP  +  VF  +  VT". 
This  law  can  easily  be  verified  by  passing  different  volumes  of 


DALTON'S  LAW.  177 

gas  into  a  graduated  glass  jar  inverted  over  mercury,  after  having 
first  measured  their  volumes  and  pressures.  It  may  be  observed  that 
Boyle's  law  is  merely  a  particular  case  of  this.  It  is  what  this  law 
becomes  when  applied  to  a  mixture  of  two  portions  of  the  same 
gas. 

,  228.  Absorption  of  Gases  by  Liquids  and  Solids. — All  gases  are  to 
a  greater  or  less  extent  soluble  in  water.  This  property  is  of  con- 
siderable importance  in  the  economy  of  nature;  thus  the  life  of 
aquatic  animals  and  plants  is  sustained  by  the  oxygen  of  the  air 
which  the  water  holds  in  solution.  The  volume  of  a  given  gas  that 
can  be  dissolved  in  water  at  a  given  temperature  is  generally  found 
to  be  approximately  the  same  at  all  pressures,1  and  the  ratio  of  this 
volume  to  that  of  the  water  which  dissolves  it  is  called  the  co- 
efficient of  solubility,  or  of  absorption.  At  the  temperature  0° 
Cent.,  the  coefficient  of  solubility  for  carbonic  acid  is  1,  for  oxygen 
•04,  and  for  ammonia  1150. 

If  a  mixture  of  two  or  more  gases  be  placed  in  contact  with  water, 
each  gas  will  be  dissolved  to  the  same  extent  as  if  it  were  the  only 
gas  present. 

Other  liquids  as  well  as  water  possess  the  power  of  absorbing 
gases,  according  to  the  same  laws,  but  with  coefficients  of  solubility 
which  are  different  for  each  liquid. 

Increase  of  temperature  diminishes  the  coefficient  of  solubility, 
which  is  reduced  to  zero  when  the  liquid  boils. 

Some  solids,  especially  charcoal,  possess  the  power  of  absorbing 
gases;  Boxwood  charcoal  absorbs  about  nine  times  its  volume  of 
oxygen,  and  about  ninety  times  its  volume  of  ammonia.  When 
saturated  with  one  gas,  if  put  into  a  different  gas,  it  gives  up  a  por- 
tion of  that  which  it  first  absorbed,  and  takes  up  in  its  place  a 
quantity  of  the  second.  Finely-divided  platinum  condenses  on  the 
surface  of  its  particles  a  large  quantity  of  many  gases,  amounting 
in  the  case  of  oxygen  to  many  times  its  own  volume.  If  a  jet  of 
hydrogen  gas  be  allowed  to  fall,  in  air,  upon  a  ball  of  spongy 
platinum,  the  gas  combines  rapidly,  in  the  pores  of  the  metal, 
with  the  oxygen  of  the  air,  giving  out  an  amount  of  heat  which 
renders  the  platinum  incandescent  and  usually  sets  fire  to  the  jet 
of  hydrogen. 

Most  solids  have  in  ordinary  circumstances  a  film  of  air  adhering 

1  Hence  the  mass  of  gas  absorbed  is  directly  as  the  pressure. 
12 


178  BOYLE'S  (OR  MARIOTTE'S)  LAW. 

to  their  surfaces.  Hence  iron  filings,  if  carefully  sprinkled  on  water, 
will  not  be  wetted,  but  will  float  on  the  surface,  and  hence  also  the 
power  which  many  insects  have  of  running  on  the  surface  of  water 
without  wetting  their  feet.  The  film  of  air  in  these  cases  prevents 
wetting,  and  hence,  by  the  principles  of  capillarity,  produces  in- 
creased buoyancy. 


CHAPTER  XX. 


AIR-PUMP. 


229.  Air-pump. — The  air-pump  was  invented  by  Otto  Guericke 
about  1G50,  and  has  since  undergone  some  improvements  in  detail 
which  have  not  altered  the  essential  parts  of  its  construction. 

Fig.  130  represents  the  pattern  most  commonly  adopted  in  France. 
It  contains  a  glass  or  metal  cylinder  called  the  barrel,  in  which 
a  piston  works.  This  piston  has  an  opening  through  it  which  is 
closed  at  the  lower  end  by  a  valve  S  opening  upwards.  The  ban-el 


Fig.  130.— Air  pump 

communicates  with  a  passage  leading  to  the  centre  of  a  brass  surface 
carefully  polished,  which  is  called  the  plate  of  the  air-pump.  The 
entrance  to  the  passage  is  closed  by  a  conical  stopper  S',  at  the  ex- 
tremity of  a  metal  rod  which  passes  through  the  piston-head  and 
works  in  it  tightly,  so  as  to  be  carried  up  and  down  with  the  motion 
of  the  piston.  A  catch  at  the  upper  part  of  the  rod  confines  its 
motion  within  very  narrow  limits,  and  only  permits  the  stopper  to 
rise  a  small  distance  above  the  opening. 


180  AIR-PUMP. 

Suppose  now  that  the  piston  is  at  the  bottom  of  the  cylinder,  and 
is  raised.  The  valve  S'  is  opened,  and  air  from  the  receiver  E  rushes 
into  the  cylinder.  On  lowering  the  piston,  the  valve  S'  closes  its 
opening,  the  air  which  has  entered  the  cylinder  cannot  return  into 
the  receiver,  and,  on  being  compressed,  raises  the  valve  S  in  the 
piston,  and  escapes  into  the  air  outside.  On  raising  the  piston 
again,  a  portion  of  the  air  remaining  in  the  receiver  will  pass  into 
the  cylinder,  whence  it  will  escape  on  pushing  down  the  piston,  and 
so  on. 

We  see,  then,  that  if  this  motion  be  continued,  a  fresh  portion  of 
the  air  in  the  receiver  will  be  removed  at  each  successive  stroke. 
But  as  the  quantity  of  air  removed  at  each  stroke  is  only  a  fraction 
of  the  quantity  which  was  in  the  receiver  at  the  beginning  of  the 
stroke,  we  can  never  produce  a  perfect  vacuum,  though  we  might 
approach  as  near  to  it  as  we  pleased  if  this  were  the  only  obstacle. 

230.  Theoretical  Rate  of  Exhaustion. — It  is  easy  to  calculate  the 
quantity  of  air  left  in  the  receiver  after  a  given  number  of  strokes 
of  the  piston.  Let  V  be  the  volume  of  the  barrel,  V  that  of  the 
receiver,  and  M  the  mass  of  air  in  the  receiver  at  first.  On  raising 
the  piston,  the  air  which  occupied  the  volume  V  occupies  a  volume 
V  +  V;  of  the  air  thus  expanded  the  volume  V  is  removed,  and  the 

volume  V  left,  being  ^p^  of  the  whole  quantity  or  mass  M.     The 
quantity  remaining  after  the  second  stroke  is  y,V'v  of  that  after  the 


first,  or  is  (v,  +  v j  M;  and  after  n  strokes  (v,V'v)nM.  Hence  the 
density  and  (by  Boyle's  law)  the  pressure  are  each  reduced  by  n 
strokes  to  (yq.-y )  of  their  original  values. 

This  calculation  gives  the  theoretical  rate  of  exhaustion  for  a 
perfect  pump.  Ordinary  pumps  come  nearly  up  to  this  standard 
during  the  earlier  part  of  the  process  of  exhaustion;  but  as  further 
progress  is  made,  the  imperfections  of  the  apparatus  become  more 
sensible,  and  set  a  limit  to  the  exhaustion  attainable. 

231.  Mercurial  Gauges.— To  enable  the  operator  to  observe  the 
progress  of  the  exhaustion,  the  instrument  is  usually  provided  with 
a  mercurial  gauge.  Sometimes,  as  in  Fig.  130,  this  consists  of  a 
short  siphon-barometer,  the  difference  of  levels  between  its  two 
columns  being  the  measure  of  the  pressure  in  the  receiver.  Another 
plan  is  to  have  a  straight  tube  open  at  both  ends,  and  moro  than  30 


RATE   OF   EXHAUSTION.  181 

inches  long;  its  upper  end  being  connected  with  the  receiver,  while 
its  lower  end  dips  into  a  cistern  of  mercury.  As  exhaustion  pro- 
ceeds, the  mercury  rises  in  this  tube,  and  its  height  above  the 
mercury  in  the  cistern  measures  the  difference  between  the  pressure 
in  the  receiver  and  that  in  the  external  air. 

232.  Admission  Stop-cock. — After  the  receiver  has  been  exhausted 
of  air,  if  it  were  required  to  raise  it  from  the  plate,  a  very  consider- 
able force  would  be  necessary,  amounting  to  as  many  times  fifteen 
pounds  as  the  base  of  the  receiver  contained  square  inches.     This 
difficulty  is  obviated  by  having  an  admission  stop-cock  R,  which  is 
shown  in  section  above.     It   is  perforated  by  a  straight  channel, 
which,  when  the  machine  is  being  worked,  forms  part  of  the  com- 
municating passage.     At  90°  from  the  extremities  of  this  channel  is 
another  opening  O,  forming  the  mouth  of  a  bent  passage,  leading  to 
the  external  air.     When  we  wish  to  admit  the  air  into  the  receiver, 
we  have  only  to  turn  the  stop-cock  so  as  to  bring  the  opening  0  to 
the  side  next  the  receiver;  if,  on  the  contrary,  we  turn  it  towards 
the  pump-barrel,  all   communication   between   the  pump  and  the 
receiver  is  stopped,  the  risk  of  air  entering  is  diminished,  and  the 
vacuum  remains  good  for  a  greater  length  of  time.     This  precaution 
is  taken  when  we  wish  to  leave  bodies  in  a  vacuum  for  a  consider- 
able time.     Another  method  is  to  employ  a  separate  plate,  which 
can  be  detached  so  as  to  leave  the  machine  available  for  other  pur- 
poses. 

233.  Double-barrelled  Air-pump. — The  machine  just  described  has 
only  a  single  pump-barrel;  air-pumps  of  this  kind  are  sometimes 
employed,  and  are  usually  worked  by  a  lever  like  a  pump-handle. 
With  this  arrangement,  it  is  evident  that  no  air  is  expelled  in  the 
down-stroke;  and  that  the  piston,  after  having  expelled  the  air  from 
the  barrel  in  the  up-stroke,  must  descend  idle  in  order  to  prepare 
for  the  next  stroke. 

Double-barrelled  pumps  are  more  frequently  used.  An  idea  of 
their  general  arrangement  may  be  formed  from  Figs.  131,  182,  and 
133.  Fig.  133  gives  the  machine  in  perspective,  Fig.  131  is  a  section 
through  the  axes  of  the  pump-barrels,  and  Fig.  132  shows  the  manner 
in  which  communication  is  established  between  the  receiver  and  the 
two  barrels.  It  will  be  observed  that  the  two  passages  from  the 
barrels  unite  in  a  single  passage  to  the  centre  of  the  plate  p. 

Two  racks  carrying  the  pistons  CC  work  with  the  pinion  P.  This 
pinion  is  turned  by  a  double-handed  lever,  which  is  moved  alter- 


182 


AIR-PUMP. 


nately  in  opposite  directions.  In  this  arrangement,  when  one  piston 
ascends  the  other  descends,  and  consequently  in  each  single  stroke 
the  air  of  the  receiver  passes  into  one  or  other  pump-barrel.  A 
vacuum  is  thus  produced  by  half  the  number  of  strokes  which  would 
be  required  with  a  single-barrelled  pump.  It  has  besides  another 
advantage,  as  compared  with  the  single-barrelled  pump  above 
described.  In  that  pump  the  force  required  to  raise  the  piston 

increases  as  the  exhaus- 
tion proceeds,  and  when 
it  is  nearly  completed 
there  is  the  resistance  of 
almost  an  atmosphere  to 
be  overcome.  In  the 


Fig.  131. 


Double-  barrelled  Air-pump. 


Fig.  132. 


double-barrelled  pump,  with  the  same  construction  of  barrel,  the 
force  opposing  the  ascent  of  one  piston  is  precisely  equal,  at  the 
beginning  of  each  stroke,  to  that  which  assists  the  descent  of 
the  other.  This  equality,  however,  exists  only  at  the  beginning 
of  the  stroke;  for  the  air  below  the  descending  piston  is  compressed, 
and  its  tension  increases  till  it  becomes  equal  to  that  of  the  atmo- 
sphere and  raises  the  piston  valve.  During  the  remainder  of  the 
stroke,  the  resistance  to  the  ascent  of  the  other  piston  is  entirely 
uncompensated,  and  up  to  this  point  the  compensation  has  been 
gradually  diminishing.  But  the  more  nearly  we  approach  to  a 
perfect  vacuum,  the  later  in  the  stroke  does  this  compensation  occur. 


DOUBLE-BARRELLED   PUMP. 


183 


The  pump,  accordingly,  becomes  easier  to  work  as  the  exhaustion 
proceeds. 

234.  Single-barrelled  Pumps  with   Double  Action. — We  do  not, 
however,  require  two  pump-barrels  in  order  to  obtain  double  action, 


Fig.  133. — Air-pump. 

as  the  same  effect  may  be  obtained  with  a  single  barrel.  An  arrange- 
ment for  this  purpose  was  long  ago  suggested  by  Delahire  for  water- 
pumps;  but  the  principle  has  only  lately  been  applied  to  the  con- 
struction of  air-pumps. 

Fig.  134  represents  the  single  barrel  of  the  double-acting  pump  of 
Bianchi.  It  will  be  seen  that  the  piston-valve  opens  into  the  hollow 
piston-rod;  a  second  valve,  also  opening  upwards,  is  placed  at  the 
top  of  the  pump-barrel.  Two  other  openings,  one  above,  the  other 
below,  serve  to  establish  communication,  by  means  of  a  bent  vertical 
tube,  between  the  pump-barrel  and  the  passage  to  the  plate.  These 
openings  are  closed  alternately  by  two  conical  stoppers  at  the  two 
extremities  of  a  metal  rod  passing  through  the  piston,  and  carried 
with  it  in  its  vertical  movement  by  means  of  friction.  When  the 


184 


AIR-PUMP. 


piston  ascends,  as  in  the  figure,  the  upper  opening  is  closed  and  the 
lower  one  is  open;  when  the  piston  begins  to  descend,  the  opposite 
effect  is  immediately  produced.  Accordingly  we  see  that,  whichever 
be  the  direction  in  which  the  piston  is  moving,  the  receiver  is  being 
exhausted  of  air.  In  fact,  when  the  pis- 
ton ascends,  air  from  the  receiver  will 
enter  by  the  lower  opening,  and  the  air 
above  the  piston  will  be  gradually  com- 
pressed, and  will  finally  escape  by  the 
valve  above.  In  the  descending  move- 
ment, air  will  enter  by  the  upper  opening, 
and  the  compressed  air  beneath  the  piston 
will  escape  by  the  piston-valve.  The 
movement  of  the  piston  is  produced  by  a 
peculiar  arrangement  shown  in  Fig.  135, 
which  gives  a  general  view  of  the  ap- 
paratus. 

The   pump-barrel,  which  is  composed 
entirely  of  cast-iron,  oscillates  about  an 
axis  passing  through  its  base.      On  the 
top  are  guides  in  which  the  end  of  a 
crank   travels.     The   pump   is   worked   by  turning   a   heavy  fly- 
wheel of  cast-iron,  on  the  axis  of  which  is  a  pinion  which  drives 
a  toothed   wheel  on  the   axis   of  the 
crank.    The  end  of  the  crank  is  attached 
to  the  extremity  of  the  piston-rod.     It 
is  evident  that  on  turning  the  fly-wheel 
the  pump-barrel  will  oscillate  from  side 
to  side,  following  the  motions  of   the 
crank,  and  the  piston  will  alternately 
ascend  and  descend   in  the  barrel,  the 
length  of  which  should  be  equal  to  the 
diameter  of  the  circle  described  by  the 
end  of  the  crank. 

235.  English  forms  of  Air-pump. — 
Some  of  the  drawbacks  to  the  single- 
barrelled  pump  are  obviated  by  inserting 
a  valve,  opening  upwards,  in  the  top  of  the  barrel  as  at  U,  Fin-.  13C. 
The  top  of  the  piston  is  thus  relieved  from  atmospheric  pressure, 
and  the  operation  of  pumping  does  not  become  more  laborious  as 


Fig.  134. 
Barrel  of  Biaiiclii's  Air-pump. 


186  AIR-PUMP. 

the  exhaustion  proceeds,  but  less  laborious,  the  difference  being  most 
marked  when  tho  receiver  is  small. 

In  the  up-stroke,  the  piston-valve  V  keeps  shut,  and  the  air  above 
the  piston  is  pushed  out  of  the  barrel  through  the  valve  U.  In  the 
down-stroke,  U  is  kept  closed  by  the  preponderance  of  atmospheric 
pressure  outside,  and  V  opens,  allowing  the  air  to  pass  up  through 
it  as  the  piston  descends  to  the  bottom  of  the  barrel.  When  the 
exhaustion  is  far  advanced,  U  does  not  open  till  the  piston  has 
nearly  reached  the  top.  This  is  a  simple  and  good  form  of  pump. 

Another  form  very  much  in  use  in  this  country  is  the  double-act- 
ing pump  of  Professor  T.  Tate,  the  working  parts  of  which  are 
shown  in  Fig.  137.  CD  is  the  barrel;  A  and  B  are  two 
solid  pistons  rigidly  connected  by  a  rod,  and  moved  by 
the  piston-rod  AH,  which  passes  through  a  stuffing- 
box  S.  VV  are  valves  in  the  two  ends  of  the  barrel, 
both  opening  outwards,  and  II  is  a  passage  leading 
from  the  middle  of  the  cylinder  to  the  receiver.  The 
distance  between  the  extreme  faces  of  the  pistons  is 
about  fths  of  an  inch  less  than  half  the  length  of  the 
cylinder.  The  volume  of  air  expelled  at  each  single 
stroke  is  thus  about  half  the  volume  of  the  cylinder. 

This  figure  and  description  are  in  accordance  with 
the  original  account  of  the  pump  given  by  the  inventor 
in  the  Philosophical  Magazine.  It  is  now  usual  to  replace  the  two 
pistons  by  a  single  piston  of  great  thickness,  its  two  faces  being  as 
far  apart  as  the  extreme  faces  of  the  two  pistons  in  the  figure.  It 
is  also  usual  to  make  the  barrel  horizontal. 

The  valves  of  these  pumps,  and  of  most  English  pumps  are  "  silk 
valves."  They  consist  of  a  short  and  narrow  slit  in  a  thin  plate  of 
brass,  with  a  flap  of  oiled  silk  secured  at  both  ends  to  the  plate,  in 
such  a  position  that  its  central  portion  covers  the  slit.  When  the 
pressure  of  the  air  is  greater  on  the  further  side  of  the  plate  than 
on  the  side  where  the  silk  is,  the  flap  is  slightly  lifted  and  the  air 
gets  through;  but  excess  of  pressure  on  the  near  side  presses  the  flap 
down  over  the  slit  and  makes  it  air-tight. 

236.  Various  Experiments  with  the  Air-pump.— At  the  time  when 
the  air-pump  was  invented,  several  experiments  were  devised  to 
show  the  effects  of  a  vacuum,  some  of  which  have  become  classical, 
and  .are  usually  repeated  in  courses  of  experimental  physics. 

Burst  Bladder.— On  the  plate   of   an   air-pump   (Fig.   138)   is 


EXPERIMENTS. 


187 


placed  a  glass  cylinder  open  at  the  bottom,  and  having  a  piece  of 
bladder  or  thin  indian-rubber  tightly  stretched  over  the  top.  As 
the  exhaustion  proceeds,  this  bends  inwards  in  consequence  of  the 
atmospheric  pressure  above  it,  and  finally  bursts  with  a  loud  report. 

Magdeburg  Hemispheres. — We  take  two  hemispheres  (Fig.  139), 
which  can  be  exactly  fitted  on  each  other;  their  exact  adjustment 
is  further  assisted  by  a 
projecting  internal  rim, 
which  is  smeared  with 
lard.  The  apparatus  is 
exhausted  of  air  through 
the  medium  of  the  stop- 
cock attached  to  one  of  the 
hemispheres ;  and  when 
a  vacuum  has  been  pro- 
duced, it  will  be  found 
that  a  considerable  force 
is  required  to  separate 
the  two  parts,  this  force 
increasing  with  the  size 
of  the  hemispheres. 

This  resistance  to  sep- 
aration is  due  to  the  normal  exterior  pressure  of  the  air  on  every 
point  of  the  surface,  a  pressure  which  is  counterbalanced  by  only 
a  very  feeble  pressure  from  the  interior.  In  order  to  estimate 
the  resultant  effect  of  these  different  pressures,  let  us  suppose 
that  one  hemisphere  is  vertically  over  the  other,  and  that  the 
external  surface  is  cut  into  a  series  of  steps, — that  is  to  say,  of 
alternate  vertical  and  horizontal  elements.  It  is  evident  that  the 
pressure  urging  either  hemisphere  towards  the  other  will  be  simply 
the  sum  of  the  pressures  upon  its  horizontal  elements;  and  this  sum 
is  identical  with  the  pressure  which  would  be  exerted  upon  a  cir- 
cular area  equal  to  the  common  base  of  the  hemispheres.  For 
example,  if  this  area  is  10  square  inches,  and  the  external  pressure 
exceeds  the  internal  by  14  Ibs.  to  the  inch,  the  hemispheres  will  bo 
pressed  together  with  a  force  of  140  Ibs. 

Fountain  in  Vacuo. — The  apparatus  for  this  experiment  consists 
of  a  bell-shaped  vessel  of  glass  (Fig.  140),  the  base  of  which  is  pierced 
by  a  tube  fitted  with  a  stop-cock  which  enables  us  to  exhaust  the 
vessel  of  air,  If,  after  a  vacuum  has  been  produced,  we  place  the 


Fig. 
Magdeburg  Hemisphere. 


188 


AIR-PUMP. 


lower  end  of  the  tube  in  a  vessel  of  water,  and  open  the  stop-cock, 
the  liquid,  being  pressed  externally  by  the  atmosphere,  mounts  up 
the  tube  and  ascends  in  a  jet  into  the  interior  of  the  vessel  This 
experiment  is  often  made  in  the  opposite  manner.  Under  the 
receiver  of  the  air-pump  is  placed  a  vial  partly  filled  with  water, 

and  having  its  cork 
pierced    by    a    tube 
open   at    both    ends, 
the  lower  end  being 
beneath   the  surface 
of  the  water.     As  the 
exhaustion   proceeds, 
the  air  in  the  vial, 
by  its  excess  of  pres- 
sure,  acts   upon   the 
liquid  and  makes  it 
issue  in  a  jet. 
s    237.  Limit  to  the 
Action    of   the    Air- 
pump. — We  have  said 
above  (§230)  that  the 
air-pump     does    not 
continue  the  process 
of  rarefaction  indefi- 
nitely, but  that  at  a 
certain  stage  its  effect 
ceases,  and   the  pressure  of  the  air  in  the  receiver  undergoes  no 
further  diminution.     If  the  pump  is  very  badly  made,  this  pres- 
sure is  considerable;  but  even  with  the  most  perfect  machines  it  is 
always  sensible.     A  pump  such  as  we  have  described  may  be  con- 
sidered good  if  it  reduces  the  pressure  of  the  air  in  the  receiver  to 
a  tenth  of  an  inch  of  mercury.     A  fiftieth  of  an  inch  is  perhaps  the 
lowest  limit. 

LEAKAGE. — This  limit  to  the  action  of  the  machine  is  due  to  vari- 
ous causes.  In  the  first  place,  there  is  frequently  leakage  at  different 
parts  of  the  apparatus;  and  although  at  the  beginning  of  the  opera- 
tion the  quantity  of  air  which  thus  enters  is  small  in  comparison 
with  that  which  is  pumped  out,  still,  as  the  exhaustion  proceeds,  the 
air  enters  faster,  on  account  of  the  diminished  internal  pressure,  and 
at  the  same  time  the  quantity  expelled  at  each  stroke  becomes  less, 


LIMITS   TO   ACTION.  189 

so  that  at  length  a  point  is  reached  at  which  the  inflow  and  outflow 
are  equal. 

In  order  to  prevent  leakage  as  far  as  possible,  the  plate  of  the 
pump  and  the  base  of  the  receiver  must  be  truly  plane  so  as  to  fit 
accurately;  the  base  of  the  receiver  must  be  ground  (that  is  rough- 
ened) and  must  be  well  greased  before  pressing  it  down  on  the  plate. 
The  piston  must  also  be  well  lubricated  with  oil. 

SPACE  UXTRAVERSED  BY  PISTON. — Another  reason  of  imperfect 
exhaustion  is  that,  after  all  possible  precautions,  a  space  is  still  left 
between  the  bottom  of  the  pump-barrel  and  the  lower  surface  of  the 
piston  when  the  latter  is  at  the  end  of  its  downward  stroke.  It  is 
evident  that  at  this  moment  the  air  contained  in  this  untraversed 
space  is  of  the  same  tension  as  the  atmosphere.  On  raising  the 
piston,  this  air  is  indeed  rarefied;  but  it  still  preserves  a  certain 
tension,  and  it  is  evident  that  when  the  air  in  the  receiver  has  been 
brought  to  this  stage  of  rarefaction,  the  machine  will  cease  to  pro- 
duce any  effect. 

If  v  is  the  volume  of  this  space,  and  V  the  volume  of  the  pump- 
barrel,  the  air,  which  at  volume  v  has  a  pressure  H  equal  to  that  of 

the  atmosphere,  will  have,  at  volume  V,  a  pressure  H  ^-     This  gives 

the  limit  to  the  action  of  the  machine  as  deduced  from  the  consider- 
ation of  the  untraversed  space. 

AIR  GIVEN  OUT  BY  OIL. — Finally,  perhaps  the  most  important 
cause,  and  the  most  difficult  to  remedy,  is  the  absorption  of  air  by 
the  oil  used  for  lubricating  the  pistons.  This  oil  is  poured  on  the 
top  of  the  piston,  but  the  pressure  of  the  external  air  forces  it  be- 
tween the  piston  and  the  barrel,  whence  it  falls  in  greater  or  less 
quantity  to  the  bottom  of  the  barrel,  where  it  absorbs  air,  and  par- 
tially yields  it  up  at  the  moment  when  the  piston  begins  to  rise, 
thus  evidently  tending  to  derange  the  working  of  the  machine.  It 
has  been  attempted  to  get  rid  of  untraversed  space  by  employing  a 
kind  of  piston  of  mercury.  This  has  also  the  advantage  of  fitting 
the  barrel  more  accurately,  and  thus  preventing  the  entrance  of  air. 
The  use  of  oil  is  at  the  same  time  avoided,  and  we  thus  escape  the 
injurious  effects  mentioned  above.  We  proceed  to  describe  two 
machines  founded  upon  this  principle. 

238.  Kravogl's  Air-pump. — This  contains  a  hollow  glass  cylinder 
AB  (Fig.  141)  tapering  at  the  upper  end,  and  surmounted  by  a  kind 
of  funnel.  The  piston  is  of  the  same  shape  as  the  cylinder,  and  is 


190 


AIR-PUMP. 


covered  with  a  layer  o£  mercury,  whose  depth  over  the  point  of  the 
piston  isaboutAthof  an  inch  when  the  piston  ,s  at  the  bottom  of  ts 
stroke,  but  is  nearly  an  inch  when  the  piston  rises  and  fills  the 


Fig.  141.— Kravogl's  Air-pump. 


funnel-shaped  cavity  in  which  the  pump-barrel  terminates.  A 
small  interval,  filled  by  the  liquid,  is  left  between  the  barrel  and  the 
piston;  but  at  the  bottom  of  the  barrel  the  piston  passes  through 
a  leather  box  carefully  made,  so  as  to  be  perfectly  air-tight. 

The  air  from  the  receiver  enters  through  the  lateral  opening  e,  and 


KRAVOGL'S  PUMP.  191 

is  driven  before  the  mercury  into  the  funnel  above.  With  the  air 
passes  a  certain  quantity  of  mercury,  which  is  detained  by  a  steel 
valve  c  at  the  narrowest  part  of  the  funnel.  This  valve  rises  auto- 
matically when  the  surface  of  the  mercury  is  at  a  distance  of  about 
half  an  inch  from  the  funnel,  and  falls  back  into  its  former  position 
when  the  piston  is  at  the  end  of  its  upward  stroke.  In  the  down- 
ward stroke,  when  the  mercury  is  again  half  an  inch  from  the  funnel, 
the  valve  opens  again  and  allows  a  portion  of  the  mercury  to  pass. 

The  effect  of  this  arrangement  is  easily  understood;  there  is  no 
"  untraversed  space,"  the  presence  of  the  mercury  above  and  around 
the  piston  causes  a  very  complete  fit,  and  excludes  the  external  air; 
and  hence  the  machine,  when  well  made,  is  very  effective. 

When  this  is  the  case,  and  when  the  mercury  used  in  the  apparatus 
is  perfectly  dry,  a  vacuum  of  about  ^i^th  of  an  inch  can  be  obtained. 
The  dryness  of  the  mercury  is  a  very  important  condition,  for  at 
ordinary  temperatures  the  elastic  force  of  the  vapour  of  water  has  a 
very  sensible  value.  If  we  wish  to  employ  the  full  powers  of  the 
machine,  we  must  have,  between  the  vessel  to  be  exhausted  of  air 
and  the  pump-barrel,  a  desiccating  apparatus. 

The  arrangement  of  the  valve  e  is  peculiar.  It  is  of  a  conical 
form,  so  as,  in  its  lowest  position,  to  permit  the  passage  of  air  coming 
from  the  receiver.  Its  ascent  is  produced  by  the  pressure  of  the 
mercury,  which  forces  it  against  the  conical  extremity  of  the  passage, 
and  the  liquid  is  thus  prevented  from  escaping. 

The  figure  represents  a  double-barrelled  machine  analogous  to  the 
ordinary  air-pump.  Besides  the  pinion  working  with  the  racks  of 
the  pistons,  there  is  a  second  smaller  pinion,  not  shown  in  the  figure, 
which  governs  the  movements  of  the  valves  c.  All  the  parts  of  this 
machine,  as  the  stop-cocks,  valves,  pipes,  &c.,  must  be  of  steel,  to 
avoid  the  action  which  the  mercury  would  have  upon  any  other 
metal. 

239.  Geissler's  Machine. — Geissler,  of  Bonn,  invented  a  mercurial 
air-pump,  in  which  the  vacuum  is  produced  by  communication 
of  the  receiver  with  a  Torricellian  vacuum.  Fig.  142  represents 
this  machine  as  constructed  by  Alvergniat.  It  consists  of  a  vertical 
tube,  serving  as  a  barometric  tube,  and  communicating  at  the  bottom, 
by  means  of  a  caoutchouc  tube,  with  a  globe  which  serves  as  the 
cistern. 

At  the  top  of  the  tube  is  a  three-way  stop-cock,  by  which  com- 
munication can  be  established  either  with  the  receiver  to  the  left,  or 


192 


AIR-PUMP. 


with  a  funnel  to  the  right,  which  latter  has  an  ordinary  stop-cock 
at  the  bottom.  By  means  of  another  stop-cock  on  the  lett,  com- 
munication with  the  receiver  can  be  opened  or  closed.  These  stop- 
cocks are  made  entirely 
of  glass.  The  machine 
works  in  the  following- 
manner;  communication 
being  established  with 
the  funnel,  the  globe 
which  serves  as  cistern 
is  raised,  and  placed,  as 
shown  in  the  figure,  at 
a  higher  level  than  the 
stop-cock  of  the  funnel. 
By  the  law  of  equili- 
brium in  communicat- 
ing vessels,  the  mercury 
fills  the  barometric  tube, 
the  neck  of  the  funnel, 
and  part  of  the  funnel 
itself.  If  the  communi- 
cation between  the  fun- 
nel and  tube  be  now 
stopped,  and  the  globe 
lowered,  a  Torricellian 
vacuum  is  produced  in 
the  upper  part  of  the 
vertical  tube. 

Communication  is  now 

---    j^T1— ^^gi^^Hmr-^^^-^^    opened    with    the    re- 
i^^BMUl^^!^  =    ceiver;    the   air   rushes 

into  the  vacuum, and  the 
column  of  mercury  falls 

rig.  ui-Geissier-s  Machine.  a  little.  Communication 

is  now  stopped  between 

the  tube  and  receiver,  and  opened  between  the  tube  and  the  funnel, 
the  simple  stop-cock  of  the  funnel  being,  however,  left  shut.  If  at 
this  moment  the  globe  is  replaced  in  the  position  shown  in  the 
figure,  the  air  tends  to  escape  by  the  funnel,  and  it  is  easy  to  allow 
it  to  do  so.  Thus,  a  part  of  the  air  of  the  receiver  has  been  removed, 


GEISSLER'S  AND  SPRENGEL'S.  193 

and  the  apparatus  is  in  the  same  position  as  at  the  beginning.  The 
operation  described  is  equivalent  to  a  stroke  of  the  piston  in  the 
ordinary  machine,  and  this  process  must  be  repeated  till  the  receiver 
is  exhausted. 

As  the  only  mechanical  parts  of  this  machine  are  glass  stop-cocks, 
which  are  now  executed  with  great  perfection,  it  is  capable  of  giving 
very  good  results.  With  dry  mercury  a  vacuum  of  -^-jrth  of  an  inch 
may  very  easily  be  obtained.  The  working  of  the  machine,  how- 
ever, is  inconvenient,  and  becomes  exceedingly  laborious  when  the 
receiver  is  large.  It  is  therefore  employed  directly  only  for  pro- 
ducing a  vacuum  in  very  small  vessels;  when  the  spaces  to  be 
exhausted  of  air  are  at  all  large,  the  operation  is  begun  with  the 
ordinary  machine,  and  the  mercurial  air-pump  is  only  employed  to 
render  the  vacuum  thus  obtained  more  perfect. 

240.  Sprengel's  Air-pump. — This  instrument,  which  may  be  re- 
garded as  an  improvement  upon  Geissler's,  is  represented  in  its 
simplest  form  in  Fig.  143.  cd  is  a  glass  tube  longer  than  a  baro- 
meter tube,  down  which  mercury  is  allowed  to  fall  from  the  funnel 
A.  Its  lower  end  dips  into  the  glass  vessel  B,  into  which  it  is  fixed 
by  means  of  a  cork.  This  vessel  has  a  spout  at  its  side,  a  few  milli- 
metres higher  than  the  lower  end  of  the  tube.  The  first  portions 
of  mercury  which  run  down  will  consequently  close  the  tube,  and 
prevent  the  possibility  of  air  entering  it  from  below.  The  upper 
part  of  cd  branches  off  at  x  into  a  lateral  tube  communicating  with 
the  receiver  B.,  which  it  is  required  to  exhaust.  A  convenient 
height  for  the  whole  instrument  is  6  feet.  The  funnel  A  is 
supported  by  a  ring  as  shown  in  the  figure,  or  by  a  board  with  a 
hole  cut  in  it.  The  tube  cd  consists  of  two  parts,  connected  by  a 
piece  of  india-rubber  tubing,  which  can  be  compressed  by  a  clamp 
so  as  to  keep  the  tube  closed  when  desired.  As  soon  as  the  mercury 
is  allowed  to  run  down,  the  exhaustion  begins,  and  the  whole  length 
of  the  tube,  from  x  to  d,  is  seen  to  be  filled  with  cylinders  of  mercuiy 
separated  by  cylinders  of  air,  all  moving  downwards.  Air  and 
mercury  escape  through  the  spout  of  the  bulb  B,  which  is  above  the 
basin  H,  where  the  mercury  is  collected.  This  has  to  be  poured 
back  from  time  to  time  into  the  funnel  A,  to  pass  through  the  tube 
again  and  again  until  the  exhaustion  is  completed. 

As  the  exhaustion  is  progressing,  it  will  be  noticed  that  the  inclosed 
air  between  the  mercury  cylinders  becomes  less  and  less,  until  the 
lower  part  of  cd  presents  the  aspect  of  a  continuous  column  of  mer- 
13 


194 


AIR-PUMP. 


cury  about  30  inches  high.  Towards  this  stage  of  the  operation  a 
considerable  noise  begins  to  be  heard,  similar  to  that  of  a  shaken 
water-hammer,  and  common  to  all  liquids  shaken  in  a  vacuum.  The 
operation  may  be  considered  completed  when  the  column  of  mercury 
does  not  inclose  any  air,  and  when  a  drop  of  mercury  falls  upon  the 
top  of  this  column  without  inclosing  the  slightest  air-bubble.  The 

height  of  this  column  now 
corresponds  exactly  with  the 
height  of  the  column  of  mer- 
cury in  a  barometer;  or,  what 
is  the  same,  it  represents  a 
barometer  whose  vacuum  is 
the  receiver  E,  and  connecting 
tube. 

Dr.  Sprengel  recommends  the 
employment  of  an  auxiliary 
air-pump  of  the  ordinary  kind, 
to  commence  the  exhaustion, 
when  time  is  an  object,  as  with- 
out this  from  20  to  30  minutes 
are  required  to  exhaust  a 
receiver  of  the  capacity  of  half 
a  litre.  As,  however,  the  em- 
ployment of  the  auxiliarypump 
involves  additional  connections 
and  increased  leakage,  it  should 
be  avoided  when  the  best  pos- 
sible exhaustion  is  desired.  The 
fall  tube  must  not  exceed  about 
a  tenth  of  an  inch  in  diameter, 
and  special  precautions  must 
be  employed  to  make  the  india- 
rubber  connections  air-tight.  (See  Chemical  Journal  for  18G5,  p.  9.) 
By  this  instrument  air  has  been  reduced  to  iaouoooth  of  atmo- 
spheric density,  and  the  average  exhaustion  attainable  by  its  use  is 
about  one-millionth,  which  is  equivalent  to  '00003  of  an  inch  of 
mercury. 

241.  Double  Exhaustion.— In  the  mercurial  machines  just  described 
fcfieraia  no  "untraversed  space,"  as  the  liquid  completely  expels  all 
the  air  from  the  pump-barrel.  These  machines  are  of  Very  recent 


Fig.  143.— Sprengel's  Air-pump. 


DOUBLE    EXHAUSTION. 


195 


invention.  Babinet  long  before  introduced  an  arrangement  for  the 
purpose,  not  of  getting  rid  of  this  space,  but  of  exhausting  it  of  air. 

For  this  purpose,  when  the  machine  ceases  to  work  with  the  ordi- 
nary arrangement,  the  communication  of  the  receiver  with  one  of 
the  pump-barrels  is  shut  off,  and  this  barrel  is  employed  to  exhaust 
the  air  from  the  other.  This  change  is  effected  by  means  of  a  stop- 
cock at  the  point  of  junction  of  the  passages  leading  from  the  two 
barrels  (Fig.  144).  The  stop-cock  has  a  T-shaped  aperture,  the  point 
of  intersection  of  the  two  branches  being  in  constant  communication 
with  the  receiver.  In  a  dif- 
ferent plane  from  that  of  the 
T-shaped  aperture  is  another 
aperture  mn,  which,  by  means 
of  the  tube  I,  'establishes 
communication  between  the 
pump-barrel  B  and  the  com- 
municating passage  of  the 
pump-barrel  A.  From  this 
explanation  it  will  be  seen  that 
if  the  stop-cock  be  turned  as 
shown  in  the  first  figure,  the 
two  pump-barrels  both  com- 
municate with  the  receiver, 
and  the  operation  proceeds  in 
the  ordinary  manner.  But  if 
the  stop-cock  be  turned  through 

a  quarter  of  a  revolution,  as  shown  in  the  second  figure,  the  pump- 
barrel  B  alone  communicates  with  the  receiver,  while  it  is  itself 
exhausted  of  air  by  the  barrel  A. 

It  is  easy  to  express  by  a  formula  the  effect  of  this  double  exhaus- 
tion. Suppose  the  pump  to  have  ceased,  under  the  ordinary  method 
of  working,  to  produce  any  farther  exhaustion,  the  air  in  the  receiver 

has  therefore  reached  a  tension  nearly  equal  to  H^.  (§  237).     At  this 

moment  the  stop-cock  is  turned  into  its  second  position.  When  the 
piston  B  descends,  the  piston  A  rises,  and  the  air  of  the  "untraversed 
space "  in  B  is  drawn  into  A  and  rarefied.  During  the  inverse 
operation,  the  air  in  A  is  prevented  from  returning  to  B,  and  thus 
the  rarefied  air  from  B,  becoming  still  further  rarefied,  will  draw  a 
fresh  quantity  of  air  from  the  receiver.  This  air  will  then  be  driven 


Fig.  144.— Babinefs  Doubly-exhausting  Stop-cock. 


196  AIR-PUMP. 

into  A,  where  it  will  be  compressed  by  the  descending  movement  of 
the  piston,  and  will  find  its  way  into  the  air  outside.1 

This  double  exhaustion  will  itself  cease  to  work  when  air  ceases  to 
pass  from  the  pump-barrel  B  into  the  pump-barrel  A.  Now  when 
the  piston  in  this  latter  is  raised,  the  elastic  force  of  the  air  which 
was  contained  in  its  "  untraversed  space  "  is  equal  to  H^,  for,  on  the 
last  opening  of  the  valve,  the  air  in  this  space  escaped  into  the  atmo- 
sphere. On  the  other  hand,  when  the  piston  in  B  is  at  the  end  of 
its  upward  stroke,  the  tension  of  the  air  is  the  same  as  in  the  receiver. 
Let  this  be  denoted  by  x.  When  the  piston  in  B  descends,  the  air  is 
compressed  into  the  "  untraversed  space  "  and  the  passage  leading  to 
A.  Let  the  volume  of  this  passage  be  I  Then  the  tension  will 
increase,  and  become  aj^rj-  When  the  machine  ceases  to  produce 
any  farther  effect,  this  tension  cannot  be  greater  than  that  in  the 
pump-barrel  A,  which  is  H|.;  we  have  thus,  to  determine  the  limit 
to  the  action  of  the  pump,  the  equation 


whence 


242.  Air-pump  with  Free  Piston.  —  We  shall  describe  one  more 
air-pump  (Fig.  145),  constructed  by  Deleuil,  and  founded  upon  an 
interesting  principle.  We  know  that  gases  possess  a  remarkable 
power  of  adhesion  for  solids,  so  that  a  body  placed  in  the  atmo- 
sphere may  be  considered  as  covered  with  a  very  thin  coat  of  air, 
forming,  so  to  speak,  a  permanent  envelope.  On  account  of  this  cir- 
cumstance, gases  find  very  great  difficulty  in  moving  in  very  narrow 
spaces.  This  is  the  principle  of  the  "  air-pump  with  free  piston." 

The  piston  P  (Fig.  14G),  which  is  composed  entirely  of  metal,  is  of 
considerable  length;  and  on  its  outer  surface  is  a  series  of  parallel 
circular  grooves  very  close  together.  It  does  not  touch  the  pump- 
barrel  at  any  point;  but  the  distance  between  the  two  is  very  small, 
about  '001  of  an  inch.  This  free  piston  is  surrounded  by  a  cushion 
of  air,  which  forms  its  only  stuffing,  and  is  sufficient  to  enable  the 
machine  to  work  in  the  ordinary  manner,  notwithstanding  the  per- 

1  It  will  be  observed  that  during  the  process  of  double  exhaustion  the  piston  of  B  be- 
haves like  a  solid  piston  ;  its  valve  never  opens,  because  the  pressure  below  it  is  always 
less  than  atmospheric. 


•f 


198 


AIR-PUMP. 


manent  communication  between  the  upper  and  lower  surfaces  of  the 
piston  This  machine  gives  a  vacuum  about  as  good  as  is  obtainable 
by  ordinary  pumps,  and  it  has  the  important  advantages  of  not 

requiring  oil,  and  of  having  less 
friction.  It  consequently  wears 
better,  and  is  less  liable  to  the 
development  of  heat,  which  is  a 
frequent  source  of  annoyance  in 
air-pumps.  It  is  single-barrelled 
with  double  action,  like  Bianchi's. 
The  two  openings  S  and  S'  are 
to  admit  air  from  the  receiver; 
they  are  closed  and  opened  alter- 
nately by  conical  stoppers  at  the 
end  of  the  rod  T,  which  passes 
through  the  piston,  and  is  carried 
with  it  by  friction  in  its  move- 
ment. They  communicate  with 
tubes  which  unite,  at  R',  with  a 
tube  leading  from  the  receiver. 
A  and  A'  are  valves  for  the  expul- 
sion of  the  air,  which  escapes  by 
tubes  uniting  at  R.  The  alternate 
movement  of  the  piston  is  produced 
by  what  is  called  Delahire's  gearing. 
This  depends  on  the  principle,  that 
when  a  circle  rolls  without  sliding 
in  the  interior  of  another  circle  of 
double  the  diameter,  any  point  on 
the  circumference  of  the  rolling 
circle  describes  a  diameter  of  the  fixed  circle.  In  order  to  utilize 
this  property,  the  end  of  the  piston-rod  is  jointed  to  the  extremity 
of  a  piece  of  metal  which  is  rigidly  attached  to  the  pinion  P,  the 
joint  being  exactly  opposite  the  circumference  of  the  pinion.  This 
latter  is  driven  by  a  fly-wheel  with  suitable  gearing,  and  works 
with  the  fixed  wheel  E,  which  is  toothed  on  the  inside.  Thus  the 
piston  will  freely,  and  without  any  lateral  effort,  describe  a  vertical 
line,  the  length  of  the  stroke  being  equal  to  the  diameter  of  the  fixed 
wheel. 

243.  Compressing  Pump.— It  can  easily  be  seen  from  the  descrip- 


Piston  and  Barrel  of  Deleuil's  Air-pump. 


COMPRESSING   PUMP. 


199 


tion  of  the  air-pump,  that  if  the  expulsion-valves  were  connected 
with  a  tube  communicating  with  a  reservoir,  the  air  removed  by  the 
pump  would  be  forced 
into  this  reservoir.  This 
communication  is  estab- 
lished in  the  instrument 
just  described.  If,  there- 
fore, B/  be  made  to  com- 
municate with  the  exter- 
nal air,  this  air  will  be 
continually  drawn  in  at 
that  point  and  forced  out 
into  the  reservoir  con- 
nected with  R,  so  that  the 
instrument  will  act  as  a  compress- 
ing pump.  The  compressing-pump 
is  thus  seen  to  be  the  same  instru- 
ment as  the  air-pump,  the  only 
difference  being  that  the  receiver 
is  connected  with  the  expulsion  valves,  instead  of 
with  the  exhaustion- valves;  it  is  thus,  so  to  speak, 
the  air-pump  reversed.  This  fact  can  be  very  well 
seen  in  the  structure  of  a  small  pump  frequently 
employed  in  the  laboratory,  and  represented  in 
Fig.  147. 

At  the  bottom  of  the  pump-barrel  are  two  valves, 
communicating  with  two  separate  reservoirs,  that 
on  the  left  being  an  admission-valve,  and  that  on 
the  right  an  expulsion- valve. 

When  the  piston  is  raised,  rarefaction  is  produced  in  the  reservoir 
to  the  left;  and  when  it  is  pushed  down,  the  air  in  the  reservoir 
to  the  right  is  compressed. 

In  Fig.  148  is  represented  a  compressing-pump  often  employed. 
At  the  bottom  of  the  pump-barrel  is  a  valve  b  opening  downward; 
in  a  lateral  tube  is  an  admission- valve  a  opening  inward.  The 
position  of  these  valves  is  shown  in  the  figure.  They  are  conical 
metal  stoppers,  fitted  with  a  rod  passing  through  a  hole  in  a  small 
plate  behind,  an  arrangement  which  prevents  the  valve  from  over- 
turning. The  rod  is  surrounded  by  a  small  spiral  spring,  which  keeps 
the  valve  pressed  against  the  opening.  If  the  lower  part  of  the 


Fig.  148. 
Condensing  Pump. 


200  AIR-PUMP. 

pump-barrel  be  screwed  upon  a  reservoir,  at  each  upward  stroke  of 
the  piston  the  barrel  will  be  filled  with  air  through  the  valve  a, 
and  at  every  downward  stroke  this  air  will  be  forced  into  the 
reservoir. 

If  the  lateral  tube  be  made  to  communicate  with  a  bladder  or 
gas-holder  filled  with  any  gas,  this  gas  will  be  forced  into  the 
reservoir,  and  compressed. 

244.  Calculation  of  the  Effect  of  the  Instrument. — The  density  of 
the  compressed  air  after  a  given  number  of  strokes  of  the  piston 
may  easily  be  calculated.  If  v  be  the  volume  of  the  pump-barrel, 
and  V  that  of  the  reservoir;  at  each  stroke  of  the  piston  there  is 
forced  into  the  reservoir  a  volume  of  air  equal  to  that  of  the  pump- 
barrel;  which  gives  a  volume  nv  at  the  end  of  n  strokes.  The  air 
in  the  reservoir,  accordingly,  which  when  at  atmospheric  pressure 
had  density  D,  and  occupied  a  volume  V  +  nv ,  will,  when  the  volume 

is  reduced  to  Vf  have  the  density  D  — ^-,  and  the  pressure  will,  by 
Boyle's  law,  be  -*nv  atmospheres. 


If  this  formula  were  rigorously  applicable  in  all  cases,  there 
would  be  no  limits  to  the  pressure  attainable,  except  those  depend- 
ing on  the  strength  of  the  reservoir  and  the  motive  power  available. 

But,  in  fact,  the  untraversed  space  left  below  the  piston,  when 
at  the  end  of  its  downward  stroke,  sets  a  limit  to  the  action  of  the 
instrument,  just  as  in  the  common  air-pump.  For  when  the  air  in 
the  barrel  is  reduced  from  the  volume  of  the  barrel  v  to  that  of  the 

untraversed  space  v,  its  tension  becomes  Hj  and  this  air  cannot 

pass  into  the  reservoir  unless  the  tension  of  the  air  in  the  reservoir 
is  less  than  this  quantity.  This  is  accordingly  the  utmost  limit  of 
compression  that  can  be  attained. 

We  must,  however,  carefully  distinguish  between  the  effects  of 
untraversed  space  in  the  air-pump  and  in  the  compression-pump. 
In  the  first  of  these  instruments  the  object  aimed  at  is  to  rarefy 
the  air  to  as  great  a  degree  as  possible,  and  untraversed  space 
must  consequently  be  regarded  as  a  defect  of  the  most  serious 
importance. 

The  object  of  the  condensing-pump,  on  the  contrary,  is  to  com- 
press the  air,  not  indefinitely,  but  up  to  a  certain  point.  Thus,  for 
instance,  one  pump  is  intended  to  give  a  compression  of  five  atmo- 
spheres, another  of  ten,  &c.  In  each  of  these  cases  the  maker 


RATE  OF   COMPRESSION. 


201 


provides  that  this  limit  shall  be  reached,  and  the  untraversed 
.space  has  no  injurious  effect  beyond  increasing  the  number  of 
strokes  required  to  produce  the  desired  amount  of  condensation. 

245.  Various  Contrivances  for  producing  Compression. — In  order  to 
expedite  the  process  of  compression,  several  pumps  such  as  we  have 


Fig.  149.— Connected  Pumps. 

described  are  combined,  which  may  be  done  in  various  ways.  Fig. 
149  represents  the  system  employed  by  Regnault  in  his  investiga- 
tions connected  with  Boyle's  law  and  the  elastic  force  of  vapour.  It 
consists  of  three  pumps,  the  piston-rods  of  which  are  jointed  to.  three 
cranks  on  a  horizontal  axle,  by  means  of  three  connecting-rods.  This 
axle,  which  carries  a  fly-wheel,  is  turned  by  means  of  one  or  two 
handles.  The  different  admission- valves  are  in  communication  with 
a  single  reservoir  in  connection  with  the  external  air,  and  the  com- 


202  AIR-PUMP. 

pressed  gas  is  forced  into  another  reservoir  which  is  in  communication 
with  the  experimental  apparatus. 

A  serious  obstacle  to  the  working  of  these  instruments  is  the  heat 
generated  by  the  compression  of  the  air,  which  expands  the  different 
parts  of  the  instrument  unequally,  and  often  renders  the  piston  so 
tight  that  it  can  scarcely  be  driven.  In  some  of  these  instruments 
which  are  employed  in  the  arts,  this  inconvenience  is  lessened  by 
keeping  the  lower  valves  covered  with  water,  which  has  the  addi- 
tional advantage  of  getting  rid  of  "untraversed  space."  In  this 
way  a  pressure  of  forty  atmospheres  may  easily  be  obtained  with 
air.  Air  may  also  be  compressed  directly,  without  the  intervention 
of  pumps,  when  a  sufficient  height  of  water  can  be  obtained.  It  is 
only  necessary  to  lead  the  liquid  in  a  tube  to  the  bottom  of  a 
reservoir  containing  air.  This  air  will  be  compressed  until  its 
pressure  exceeds  that  of  the  atmosphere  by  the  amount  due  to  the 
height  of  the  summit  of  the  tube.  It  is  by  a  contrivance  of  this 
kind  that  compressed  air  has  been  obtained  for  driving  the  boring- 
machines  employed  in  the  great  Alpine  tunnels. 

246.  Practical  Applications  of  the  Air-pump  and  of  Compressed  Air. 
— Besides  the  use  made  of  the  air-pump  and  the  compression-pump 
in  the  laboratory,  these  instruments  are  variously  employed  in  the 
arts. 

The  air-pump  is  employed  by  sugar-refiners  to  lower  the  boiling 
point  of  the  syrup.  Compression-pumps  are  used  by  soda-water 
manufacturers  to  force  the  carbonic  acid  into  the  reservoirs  contain- 
ing the  water  which  is  to  be  aerated.  The  small  apparatus  described 
above  (Fig.  148)  is  sufficient  for  this  purpose;  it  is  only  necessary 
to  fill  the  side-vessel  with  carbonic  acid,  and  to  pour  a  certain 
quantity  of  water  into  the  reservoir  below.  Compressed  air  has  for 
several  years  been  employed  to  assist  in  laying  the  foundations  of 
bridges  in  rivers  where  the  sandy  nature  of  the  soil  requires  very 
deep  excavations.  Large  tubes  called  caissons,  in  connection  with 
a  condensing  pump,  are  gradually  let  down  into  the  river;  the  air  by 
its  pressure  keeps  out  the  water,  and  the  workmen,  who  are  admitted 
into  the  apparatus  by  a  sort  of  lock,  are  thus  enabled  to  walk  on 
dry  ground.' 

In  pneumatic  despatch  tubes,  which  have  recently  been  established 
in  many  places,  a  kind  of  train  is  employed,  consisting  of  a  piston 
preceded  by  boxes  containing  the  despatches.  By  exhausting  the 
air  at  the  forward  end  of  the  tube,  or  forcing  in  compressed  air  at 


PRACTICAL  APPLICATIONS.  203 

the  other  end,  the  train  is  blown  through  the  tube  with  great 
velocity. 

The  atmospheric  railway,  which  was  for  a  few  years  in  existence, 
was  worked  upon  the  same  principle:  an  air-tight  piston  travelled 
through  a  fixed  tube,  and  was  connected  by  an  ingenious  arrange- 
ment with  a  train  above. 

Excavating  machines  driven  by  compressed  air  are  coming  into 
extensive  use  in  mining  operations.  They  have  the  advantage  of 
assisting  ventilation,  inasmuch  as  the  compressed  air,  which  at  each 
stroke  of  the  machine  escapes  into  the  air  of  the  mine,  cools  as  it 
expands. 

In  the  air-gun,  the  bullet  is  projected  by  a  portion  of  compressed 
air  which,  on  pulling  the  trigger,  escapes  into  the  barrel  from  a 
reservoir  in  which  it  has  been  artificially  compressed. 

We  may  add  that  the  large  machines  employed  in  iron- works  for 
supplying  air  to  the  furnaces,  are  really  compression-pumps. 


CHAPTEE  XXL 


UPWARD  PRESSURE   OF  THE  AIR. 


247.  The  Baroscope. — The  principle  of  Archimedes,  explained  in 
Chap.  XIII.,  applies  to  all  fluids,  whether  liquid  or  gaseous.     Hence 
the  resultant  of  the  whole  pressure  of  the  atmosphere  on  the  surface 
of  a  body  is  equal  to  the  weight  of  the  air  displaced.     The  force 
required  to  support  a  body  in  air,  is  less  than  the  force  required  to 
support  it  in  vacuo,  by  this  amount.     This  principle  is  illustrated 
by  the  baroscope  (Fig.  150). 

This  is  a  kind  of  balance,  the  beam  of  which  supports  two  balls  of 
very  unequal  sizes,  which  balance  each  other  in  the  air.  If  the  ap- 
paratus is  placed  under  the  receiver 
of  an  air-pump,  after  a  few  strokes 
of  the  piston  the  beam  will  be  seen 
to  incline  towards  the  larger  ball, 
and  the  inclination  will  increase  as 
the  exhaustion  proceeds.  The  reason 
is  that  the  air,  before  it  was  pumped 
out,  produced  an  upward  pressure, 
which  was  greater  for  the  large 
than  for  the  small  ball,  on  account 
of  its  greater  displacement;  and  this 
disturbing  force  is  now  removed. 

If  after  exhausting  the  air,  car- 
bonic acid,  which  is  heavier  than  air,  were  admitted  at  atmospheric 
pressure,  the  large  ball  would  be  subjected  to  a  greater  increase  of 
upward  pressure  than  the  small  one,  and  the  beam  would  incline  to 
the  side  of  the  latter. 

248.  Balloons.— Suppose  a  body  to  be  lighter  than  an  equal  volume 
of  air,  then  this  body  will  rise  in  the  atmosphere.     For  example,  if 


Fig.  150.— Baroscope. 


PRINCIPLE   OF   THE   BALLOOX.  205 

we  fill  soap-bubbles  with  hydrogen  (Fig.  151),  and  shake  them  off 
from  the  end  of  the  tube  at  which  they  are  formed,  they  will  be  seen, 
if  sufficiently  large,  to  ascend  in  the  air.  This  curious  experiment 
is  due  to  the  philosopher  Cavallo,  who  announced  it  in  1782.1 

The  same   principle  applies   to   balloons,  which  essentially  con- 
sist of   an  envelope  inclosing  a  gas   lighter  than  air.     In  conse- 


Fig.  151.—  Asceut  of  Soap-bubbles  filled  with  Hydrogen. 

quence  of  this  difference  of  density,  we  can  always,  by  taking  a 
sufficiently  large  volume,  make  the  weight  of  the  gas  and  containing 
envelope  less  than  that  of  the  air  displaced.  In  this  case  the  balloon 
will  ascend. 

The  invention  of  balloons  is  due  to  the  brothers  Joseph  and  Ste- 
phen Montgolfier.  The  balloons  made  by  them  were  globe-shaped, 
and  constructed  of  paper,  or  of  paper  covered  with  cloth,  the  air  in- 
side being  rarefied  by  the  action  of  heat.  It  is  curious  to  remark 

1  The  first  idea  of  a  balloon  must  be  attributed  to  Francisco  de  Lana,  who,  about 
1670,  proposed  to  exhaust  the  air  in  globes  of  copper  of  sufficient  size  and  thinness  to  weigh 
less,  under  these  conditions,  than  the  air  displaced.  The  experiment  was  not  tried,  and 
would  certainly  not  have  succeeded,  for  the  pressure  of  the  atmosphere  would  have  caused 
the  globes  to  collapse.  The  theory,  however,  was  thoroughly  understood  by  the  author, 
who  made  an  exact  calculation  of  the  amount  of  force  tending  to  make  the  globes  ascend. 
— D. 


206 


UPWARD   PRESSURE   OF  THE  AIR. 


that  in  their  first  attempts  they  employed  hydrogen  gas,  and  showed 
that  balloons  filled  with  this  gas  could  ascend.  But  as  the  hydrogen 
readily  escaped  through  the  paper,  the  flight  of  the  balloons  was 
short,  and  thus  the  use  of  hydrogen  was  abandoned,  and  hot  air  was 
alone  employed. 

The  name  montgoljteres  is  still  often  applied  to  fire-balloons.    They 
generally  consist  of  a  paper  envelope  with  a  wide  opening  below, 


Fig.  152.— Fire-balloon  of  Pilatre  cle  Rozier. 

in  the  centre  of  which  is  a  sponge  held  in  a  wire  frame.  The  sponge 
is  dipped  in  spirit  and  ignited,  when  the  balloon  is  to  be  sent  up. 

The  first  public  experiment  of  the  ascent  of  a  balloon  was  per- 
formed at  Annonay  on  the  5th  June,  1783.  On  October  21st  of  the 
same  year,  Pilatre  de  Rozier  and  the  Marquis  d'Arlandes  achieved 
the  first  aerial  voyage  in  a  fire-balloon,  represented  in  our  figure. 

Charles  proposed  to  reintroduce  the  use  of  hydrogen  by  employing 
an  envelope  less  permeable  to  the  gas.  This  is  usually  made  of  silk 
varnished  on  both  sides,  or  of  two  sheets  of  silk  with  a  sheet  of 
india-rubber  between.  Instead  of  hydrogen,  coal-gas  is  now  gener- 
ally employed,  on  account  of  its  cheapness  and  of  the  facility  with 
which  it  can  be  procured. 


BALLOONS.  207 

249. — The  lifting  power  of  a  balloon  is  the  difference  between  its 
weight  and  that  of  the  air  displaced.  It  is  easy  to  compare  the 
three  modes  of  inflation  in  this  respect. 

A  cubic  metre  of  air  weighs  about 1'300  kilogramme. 

A  cubic  metre  of  hydrogen  '089  „ 

A  cubic  metre  of  coal-gas   about  '750          „ 

A  cubic  metre  of  air  heated  to  200°  Cent. '750          „ 

We  thus  see  that  the  lifting  power  per  cubic  metre  with  hydrogen 
is  1'211,  and  with  coal-gas  or  hot  air  about  '500  kilogramme. 
If,  for  instance,  the  total  weight  to  be  raised  is  estimated  at  1500 
kilogrammes,  the  volume  of  a  balloon  filled  with  hydrogen  capable 

of  raising  the  weight  will  be  p^  =  1239  cubic  metres.     If  coal-gas 

were  employed,  the  required   volume  would  be  ?^  =  2727  cubic 

metres. 

The  car  in  which  the  aeronauts  sit  is  usually  made  of  wicker-work 
or  whalebone.  It  is  sustained  by  cords  attached  to  a  net-work 
(Fig.  153)  covering  the  entire  upper  half  of  the  balloon,  so  as  to 
distribute  the  weight  as  evenly  as  possible.  The  balloon  terminates 
below  in  a  kind  of  neck  opening  freely  into  the  air.  At  the  top 
there  is  another  opening  in  the  inside,  which  is  closed  by  a  valve 
held  to  by  a  spring.  Attached  to  the  valve  is  a  cord  which  passes 
through  the  interior  of  the  balloon,  and  hangs  above  the  car  within 
reach  of  the  hand  of  the  aeronaut. 

When  the  aeronaut  wishes  to  descend,  he  opens  the  valve  for  a 
few  moments  and  allows  some  of  the  gas  to  escape.  An  important 
part  of  the  equipment  consists  of  sand-bags  for  ballast,  which  are 
gradually  emptied  to  check  too  rapid  descent.  In  the  figure  is 
represented  a  contrivance  called  a  parachute,  by  means  of  which  the 
descent  is  sometimes  effected.  This  is  a  kind  of  large  umbrella  with 
a  hole  at  the  top,  from  the  circumference  of  which  hang  cords  sup- 
porting a  small  car.  When  the  parachute  is  left  to  itself,  it  opens 
out,  and  the  resistance  of  the  air,  acting  upon  a  large  surface, 
moderates  the  rate  of  descent.  The  hole  at  the  top  is  essential  to 
safety,  as  it  affords  a  regular  passage  for  air  which  would  otherwise 
escape  from  time  to  time  from  under  the  edge  of  the  parachute,  thus 
producing  oscillations  which  might  prove  fatal  to  the  aeronaut. 

Balloons  are  not  fully  inflated  at  the  commencement  of  the 
ascent;  but  the  inclosed  gas  expands  as  the  pressure  diminishes 
outside.  The  lifting  power  thus  remains  nearly  constant  until 


208 


UPWARD    PRESSURE   OF   THE   AIR. 


the  balloon  has  risen  so  high  as  to  be  fully  inflated.  Suppose  for 
instance,  that  the  atmospheric  pressure  is  reduced  by  one-halt, 
the  volume  of  the  balloon  will  then  be  doubled;  it  will  thus  dis- 
place a  volume  of  air  twice 
as  great  as  before,  but  of 
only  half  the  density,  so  that 
the  buoyancy  will  remain 
the  same.  This  conclusion, 
however,  is  not  quite  exact, 
because  the  solid  parts  of 
the  balloon  do  not  expand 
like  the  gas,  and  the  weight 
of  air  displaced  by  them 
accordingly  diminishes  as  the 
balloon  rises.  If  the  balloon 
continues  to  ascend  after  it 
is  completely  inflated,  its  lift- 
ing power  diminishes  rapidly, 
becoming  zero  when  a  stra- 
tum of  air  is  reached  in 
which  the  weight  of  the 
volume  displaced  is  equal  to 
that  of  the  balloon  itself. 
It  is  carried  past  this  stratum 
in  the  first  instance  in  vir- 
tue of  the  velocity  which  it 
has  acquired,  and  finally 
comes  to  rest  in  it  after  a 
number  of  oscillations. 
250.  Height  Attainable. — The  pressure  of  the  air  in  the  stratum  of 
equilibrium  can  be  calculated  as  follows: 

Let  V  be  the  volume  of  gas  which  the  balloon  can  contain  when 

fully  inflated. 
v  the  volume,  and  w  the  weight,  of  the  solid  parts,  including 

the  aeronauts  themselves. 

£  the  density  of  the  gas  at  the  standard  pressure  and  tem- 
perature, and  D  the  density  of  air  under  the  same  condi- 
tions. 

Then  if  P  denote  the  standard  pressure,  and  p  the  pressure  in  the 
stratum  of  equilibrium,  the  density  of  the  gas  when  this  stratum 


Fig.  i:3.— Balloon  with  Car  ami  Parachute. 


EFFECT   OF  AIR   ON   WEIGHTS.  209 

has  been  reached  will  be  ^3,  and  the  density  of  the  air  will  be  ^D. 

Equating  the  weight  of  the  air  displaced  to  that  of  the  floating  body, 
we  have 


whence  p  can  be  determined. 

-  251.  Effect  of  the  Air  upon  the  Weight  of  Bodies.—  The  upward 
pressure  of  the  air  impairs  the  exactness  of  weighings  obtained  even 
with  a  perfectly  true  balance,  tending,  by  the  principle  of  the  baro- 
scope, to  make  the  denser  of  two  equal  masses  preponderate.  The 
stamped  weights  used  in  weighing  are,  strictly  speaking,  standards  of 
mass,  and  will  equilibrate  any  equal  masses  in  vacuo;  but  in  air  the 
equilibrium  will  be  destroyed  by  the  greater  upward  pressure  of  the 
air  upon  the  larger  and  less  dense  body.  When  the  specific  gravities 
of  the  weights  and  of  the  body  weighed  are  known,  it  is  easy  from 
the  apparent  weight  to  deduce  the  true  weight  (that  is  to  say,  the 
mass)  of  the  body. 

Let  x  be  the  real  weight  (or  mass)  of  a  body  which  balances  a 
standard  weight  of  w  grammes  when  the  weighing  is  made  in  air. 
Let  d  be  the  density  of  the  body,  5  that  of  the  standard  weight,  and 
a  the  density  of  the  air.  Then  the  weight  of  air  displaced  by  the 
body  is  ^x,  and  the  weight  of  air  displaced  by  the  standard  weight 

is  ^w.    Hence  we  have 


5       (        /i    r 

X  =  w— —  =«}l+olT-5 

>-a 

Let  us  take,  for  instance,  a  piece  of  sulphur  whose  weight  has  been 
found  to  be  100  grammes,  the  weights  being  of  copper,  the  density 
of  which  is  8*8.     The  density  of  sulphur  is  2. 
We  have,  by  applying  the  formula, 


7o  (i  -  Fa)  !  =  100'05  grammes' 


We  see  then  that  the  difference  is  not  altogether  insensible.     It 
varies  in  sign,  as  the  formula  shows,  according  as  d  or  3  is  the 
greater.     When  the  density  of  the  body  to  be  weighed  is  less  than 
14 


210  UPWARD   PRESSURE   OF   THE  AIR. 

that  of  the  weights  used,  the  real  weight  is  greater  than  the 
apparent  weight;  if  the  contrary,  the  case  is  reversed.  If  the  body 
to  be  weighed  were  of  the  same  density  as  the  weights  used,  the  real 
and  apparent  weights  would  be  equal.  We  may  remark,  that  in 
determining  the  ratio  of  the  weights  of  two  bodies  of  the  same 
density,  by  means  of  standard  weights  which  are  all  of  one  material, 
we  need  not  concern  ourselves  with  the  effect  of  the  upward  pressure 
of  the  air;  as  the  correcting  factor,  which  has  the  same  value  for 
both  cases,  will  disappear  in  the  quotient. 


CHAPTER    XXII. 


PUMPS  FOR  LIQUIDS. 


252.  Machines  for  raising  water  have  been  known  from  very  early 
ages,  and  the  invention  of  the  common  pump  is  pretty  generally 
ascribed  to  Ctesibius,  teacher  of  the  celebrated  Hero  of  Alexandria; 
but  the  true  theory  of  its  action  was  not  understood  till  the  time  of 
Galileo  and  Torricelli. 

253.  Reason  of  the  Rising  of  Water  in  Pumps. — Suppose  we  take  a 
tube  with  a  piston  at  the  bottom  (Fig.  154),and  immerse  the  lower 
end  of  it  in  water.     The  raising  of   the 

piston  tends  to  produce  a  vacuum  below  it, 
and  the  atmospheric  pressure,  acting  upon 
the  external  surface  of  the  liquid,  compels 
it  to  rise  in  the  tube  and  follow  the  upward 
motion  of  the  piston.     This  upward  move- 
ment of  the  water  would  take  place  even 
if  some  air  were  interposed  between  the 
piston  and  the  water;  for  on  raising  the 
piston,  this  air  would  be  rarefied,  and  its 
pressure  no  longer  balancing  that  of  the 
atmosphere,  this  latter  pressure  would  cause 
the  liquid   to  ascend  in  a  column  whose 
weight,  added  to  the  pressure  of  the  air 
below  the  piston,  would  be  equal   to  the 
atmospheric  pressure.     This  is  the  principle 
on  which  water  rises  in  pumps.     These  in-  p 
struments   have  a  considerable  variety  of  T^=- 
forms,  of  which  we  shall  describe  the  most  Fig.  i54.-PnncipieofSuction-pump. 
important  types. 
y  254.  Suction-pump. — The  suction-pump  (Fig.  155)  consists  of  a 


212 


PUMPS   FOR   LIQUIDS. 


cylindrical  pump-barrel  traversed  by  a  piston,  and  communicating 
by  means  of  a  smaller  tube,  called  the  suction-tube,  with  the  water 
in  the  pump-well.  At  the  junction  of  the  pump-barrel  and  the  tube 
is  a  valve  opening  upward,  called  the  suction-valve,  and  in  the 
piston  is  an  opening  closed  by  another  valve,  also  opening  upward. 

Suppose  now  the  suction-tube  to  be  filled  with  air  at  the  atmo- 
spheric pressure,  and  the  water  consequently  to  be  at  the  same  level 
inside  the  tube  and  in  the  well.  Suppose 
the  piston  to  be  at  the  end  of  its  downward 
stroke,  and  to  be  now  raised.  This  motion 
tends  to  produce  a  vacuum  below  the  pis- 
ton, hence  the  air  contained  in  the  suc- 
tion-tube will  open  the  suction-valve,  and 
rush  into  the  pump-barrel.  The  elastic 
force  of  this  air  being  thus  diminished,  the 
atmospheric  pressure  will  cause  the  water 
to  rise  in  the  tube  to  a  height  such  that 
the  pressure  due  to  this  height,  increased 
by  the  pressure  of  the  air  inside,  will  ex- 
actly counterbalance  the  pressure  of  the 
atmosphere.  If  the  piston  now  descends, 
the  suction- valve  closes,  the  water  remains 
at  the  level  to  which  it  has  been  raised, 
and  the  air,  being  compressed  in  the  barrel, 
opens  the  piston-valve  and  escapes.  At 
the  next  stroke  of  the  piston,  the  water 
will  rise  still  further,  and  a  fresh  portion 
of  air  will  escape. 

If,  then,  the  length  of  the  suction-tube 
is  less  than  about  30  feet,  the  water  will,  after  a  certain  number 
of  strokes  of  the  piston,  be  able  to  reach  the  suction-valve  and  rise 
into  the  pump-barrel.  When  this  point  has  been  reached  the 
action  changes.  The  piston  in  its  downward  stroke  compresses 
the  air,  which  escapes  through  it,  but  the  water  also  passes 
through,  so  that  the  piston  when  at  the  bottom  of  the  pump-barrel 
will  have  above  it  all  the  water  which  has  previously  risen 
into  the  barrel.  If  the  piston  be  now  raised,  supposing  the 
total  height  to  which  it  is  raised  to  be  not  more  than  34  feet  above 
the  level  of  the  water  in  the  well,  as  should  always  be  the  case, 
the  water  will  follow  it  in  its  upward  movement,  and  will  fill  the 


Fig.  155.— Suction-pump. 


SUCTION-PUMP.  213 

pump-barrel.  In  the  downward  stroke  this  water  will  pass  up 
through  the  piston-valve,  and  in  the  following  upward  stroke  it 
will  be  discharged  at  the  spout.  A  fresh  quantity  of  water  will  by 
this  time  have  risen  into  the  pump-barrel,  and  the  same  operations 
will  be  repeated. 

We  thus  see  that  from  the  time  when  the  water  has  entered  the 
pump-barrel,  at  each  upward  stroke  of  the  piston  a  volume  of  water 
is  ejected  equal  to  the  contents  of  the  pump-barrel. 

In  order  that  the  water  may  be  able  to  rise  into  the  pump-barrel, 
the  suction- valve  must  not  be  more  than  34  feet  above  the  level  of 
the  water  in  the  well,  otherwise  the  water  would  stop  at  a  certain 
point  of  the  tube,  and  could  not  be  raised  higher  by  any  farther 
motion  of  the  piston. 

Moreover,  in  order  that  the  working  of  the  pump  may  be  such 
as  we  have  described,  that  is,  that  at  each  upward  stroke  of  the 
piston  a  quantity  of  water  may  be  removed  equal  to  the  volume  of 
the  pump-barrel,  it  is  necessary  that  the  piston  when  at  the  top  of 
its  stroke  should  not  be  more  than  34  feet  above  the  water  in  the 
well. 

255.  Effect  of  untraversed  space. — If  the  piston  does  not  descend 
to  the  bottom  of  the  barrel,  it  is  possible  that  the  water  may 
fall  short  of  rising  to  the  suction-valve,  even  though  the  total 
height  reached  by  the  piston  be  less  than  34  feet.  When  the 
piston  is  at  the  end  of  its  downward  stroke,  the  air  below  it  in 
the  barrel  is  at  atmospheric  pressure;  and  when  the  limit  of 
working  has  been  reached,  this  air  will  expand  during  the  upward 
stroke  until  it  fills  the  barrel.  Its  pressure  will  now  be  the  same  as 
that  of  the  air  in  the  top  of  the  suction-tube;  and  if  this  pressure 
be  equivalent  to  h  feet  of  water,  the  height  to  which  water  can  be 
drawn  up  will  be  only  34— A  feet. 

Example.  The  suction- valve  of  a  pump  is  at  a  height  of  27  feet 
above  the  surface  of  the  water,  and  the  piston,  the  entire  length  of 
whose  stroke  is  7*8  inches,  when  at  the  lowest  point  is  31  inches 
from  the  fixed  valve;  find  whether  the  water  will  be  able  to  rise 
into  the  pump-barrel. 

When  the  piston  is  at  the  end  of  its  downward  stroke,  the  air 
below  it  in  the  barrel  is  at  the  atmospheric  pressure;  when  the 
piston  is  raised  this  air  becomes  rarefied,  and  its  pressure,  by  Boyle's 

law,  becomes  ^  that  of  the  atmosphere;  this  pressure  can  therefore 


214  PUMPS   FOR  LIQUIDS. 

balance  a  column  of  water  whose  height  is  34  x  ^9  feet,  or  9'67 
feet.  Hence,  the  maximum  height  to  which  the  water  can  attain  is 
34  -  9-67  feet  =  24'33  feet;  and  consequently,  as  the  suction-tube 
is  27  feet  long,  the  water  will  not  rise  into  the  pump-barrel,  even 
supposing  the  pump  to  be  perfectly  free  from  leakage. 

Practically,  the  pump-barrel  should  not  be  more  than  about  25 
feet  above  the  surface  of  the  water  in  the  well;  but  the  spout  may 
be  more  than  34  feet  above  the  barrel,  as  the  water  after  rising 
above  the  piston  is  simply  pushed  up  by  the  latter,  an  operation 
which  is  independent  of  atmospheric  pressure.  Pumps  in  which 
the  spout  is  at  a  great  height  above  the  barrel  are  commonly  called 
lift-pumps,  but  they  are  not  essentially  different  from  the  suction- 
pump. 

256.  Force  necessary  to  raise  the  Piston. — The  force  which  must 
be  expended  in  order  to  raise  the  piston,  is  equal  to  the  weight  of  a 
column  of  water,  whose  base  is  the  section  of  the  piston,  and  whose 
height  is  that  to  which  the  water  is  raised.  Let  S  be  the  section  of 
the  piston,  P  the  atmospheric  pressure  upon  this  area,  h  the  height 
of  the  column  of  water  which  is  above  the  piston  in  its  present 
position,  and  h'  the  height  of  the  column  of  water  below  it;  then 
the  upper  surface  of  the  piston  is  subjected  to  a  pressure  equal  to 
P  +  S  A;  the  lower  face  is  subjected  to  a  pressure  in  the  opposite 
direction  equal  to  P  —  Sh',  and  the  entire  downward  pressure  is 
represented  by  the  difference  between  these  two,  that  is,  by  S 
(h  +  h'). 

The  same  conclusion  would  be  arrived  at  even  if  the  water  had 
not  yet  reached  the  piston.  In  this  case,  let  I  be  the  height  of  the 
column  of  water  raised;  then  the  pressure  below  the  piston  is 
P  —  S  1;  the  pressure  above  is  simply  the  atmospheric  pressure  P, 
and,  consequently,  the  difference  of  these  pressures  acts  downward, 
and  its  value  is  S  I. 

'  257.  Efficiency  of  Pumps.— From  the  results  of  last  section  it 
follows  that  the  force  required  to  raise  the  piston,  multiplied  by 
the  height  through  which  it  is  raised,  is  equal  to  the  weight  of 
water  discharged  multiplied  by  the  height  of  the  spout  above  the 
water  in  the  well.  This  is  an  illustration  of  the  principle  of  work 
(§  49).  As  this  result  has  been  obtained  from  merely  statical  con- 
siderations, and  on  the  hypothesis  of  no  friction,  it  presents  too 
favourable  a  view  of  the  actual  efficiency  of  the  pump. 


EFFICIENCY   OF   PUMPS. 


215 


Besides  the  friction  of  the  solid  parts  of  the  mechanism,  there  is 
work  wasted  in  generating  the  velocity  with  which  the  fluid,  as  a 
whole,  is  discharged  at  the  spout,  and  also  in  producing  eddies  and 
other  internal  motions  of  the  fluid.  These  eddies  are  especially  pro- 
duced at  the  sudden  enlargements  and  contractions  of  the  passages 
through  which  the  fluid  flows.  To  these  drawbacks  must  be  added 
loss  from  leakage  of  water,  and  at  the  commencement  of  the  opera- 
tion from  leakage  of  air,  through  the  valves  and  at  the  circum- 
ference of  the  piston.  In  com- 
mon household  pumps,  which  are 
generally  roughly  made,  the  effi- 
ciency may  be  as  small  as  '25  or 
•3;  that  is  to  say,  the  product  of 
the  weight  of  water  raised,  and 


Fig.  It 


Suction-pump. 


the  height  through  which  it  is  raised,  may  be  only  '25  or  '3  of  the 
work  done  in  driving  the  pump. 

In  Figs.  156  and  157  are  shown  the  means  usually  employed  for 
working  the  piston.     In  the  first  figure  the  upward  and  downward 


216 


PUMPS   FOR   LIQUIDS. 


movement  of  the  piston  is  effected  by  means  of  a  lever.  The  second 
ficrure  represents  an  arrangement  often  employed,  in  which  the 
alternate  motion  of  the  piston  is  effected  by  means  of  a  rotatory 
motion.  For  this  purpose  the  piston-rod  T  is  joined  by  means  of 
the  connecting-rod  B  to  the  crank  C  of  an  axle  turned  by  a  handle 
attached  to  the  fly-wheel  V. 

.  258.  Forcing-pump. — The  forcing-pump  consists  of  a  pump-barrel 
dipping  into  water,  and  having  at  the  bottom  a  valve  opening  up- 
ward. In  communication  with 
the  pump-barrel  is  a  side- 
tube,  with  a  valve  at  the  point 
of  junction,  opening  from  the 
barrel  into  the  tube.  A  solid 
piston  moves  up  and  down  the 
pump-barrel,  and  it  is  evident 
that  when  this  piston  is  raised, 
water  enters  the  barrel  by  the 
lower  valve,  and  that  when 
the  piston  descends,  this  water 
is  forced  into  the  side-tube. 
The  greater  the  height  of 
this  tube,  the  greater  will  be 
the  force  required  to  push  the  piston  down,  for  the  resistance  to  be 
overcome  is  the  pressure  due  to  the  column  of  water  raised. 

The  forcing-pump  most  frequently  has  a  short  suction-pipe  leading 
from  the  reservoir,  as  represented  in  Fig.  159.  In  this  case  the 
water  is  raised  from  the  reservoir  into  the  barrel  by  atmospheric 
pressure  during  the  up-stroke,  and  is  forced  from  the  barrel  into  the 
ascending  pipe  in  the  down-stroke. 

259.  Plunger. — When  the  height  to  which  the  water  is  to  be  forced 
is  very  considerable,  the  different  parts  of  the  pump  must  be  very 
strongly  made  and  fitted  together,  in  order  to  resist  the  enormous 
pressure  produced  by  the  column  of  water,  and  to  prevent  leakage. 
In  this  case  the  ordinary  piston  stuffed  with  tow  or  leather  washers 
cannot  be  used,  but  is  replaced  by  a  solid  cylinder  of  metal  called  a 
plunger.  Fig.  1GO  represents  a  section  of  a  pump  thus  constructed. 
The  plunger  is  of  smaller  section  than  the  barrel,  and  passes  through 
a  stuffing-box  in  which  it  fits  air-tight.  The  volume  of  water  which 
enters  the  barrel  at  each  up-stroke,  and  is  expelled  in  the  down- 
stroke,  is  the  same  as  the  volume  of  a  length  of  the  plunger  equal 


Fig.  158. — Forcing-pump. 


FORCE-PUMPS. 


217 


to  the  length  of  stroke;  and  the  hydrostatic  pressure  to  be  overcome 

is  proportional  to  the  section  of  the  plunger,  not  to  that  of  the 

barrel.     As   the   operation 

proceeds,  air    is    set    free 

from  the  water,  and  would 

eventually  impede   the 

working  of  the  pump  were 

it  not  permitted  to  escape. 

For  this  purpose  the  plunger 

is  pierced  with  a  narrow 

passage,   which   is   opened 

from  time  to  time  to  blow 

out  the  air. 

The  drainage  of  deep 
mines  is  usually  effected 
by  a  series  of  pumps.  The 
water  is  first  raised  by 
one  pump  to  a  reservoir, 
into  which  dips  the  suction- 
tube  of  a  second  pump, 
which  sends  the  water  up  Fig.  159.  p;g.  ico. 

-  .  ,  Suction  and  Force  Pump. 

to  a  second  reservoir,  and 

so  on.     The  piston-rods  of  the  different  pumps  are  all  joined  to  a 


Fig.  101.— Fire  engine. 

single  rod  called  the  spear,  which  receives  its  motion  from  a  steam- 
engine. 


218 


PUMPS   FOR   LIQUIDS. 


260.  Fire-engine.— The  ordinary  fire-engine  is  formed  by  the  union 
of  two  forcing-pumps  which  play  into  a  common  reservoir,  contain- 
ing in  its  upper  portion  (called  the  air-chamber)  air  compressed  by 
the  working  of  the  engine.  A  tube  dips  into  the  water  in  this 
reservoir,  and  to  the  upper  end  of  this  tube  is  screwed  the  leather 
hose  through  which  the  water  is  discharged.  The  piston-rods  are 
jointed  to  a  lever,  the  ends  of  which  are  raised  and  depressed  alter- 
nately, so  that  one  piston  is  ascending  while  the  other  is  descending. 
Water  is  thus  continually  being  forced  into  the  common  reservoir 
except  at  the  instant  of  reversing  stroke,  and  as  the  compressed  air 
in  the  air-chamber  performs  the  part  of  a  reser- 
voir of  work  (nearly  analogous  to  the  fly-wheel), 
the  discharge  of  water  from  the  nozzle  of  the 
hose  is  very  steady. 

The  engine  is  sometimes  supplied  with  water 
by  means  of  an  attached  cistern  (as  in  Fig.  162) 
into  which  water  is  poured;  but  it  is  more 
usually  furnished  with  a  suction-pipe  which 
renders  it  self -feeding. 

261.  Double-acting  Pumps. — These  pumps,  the 
invention  of  which  is  due  to  Delahire,  are  often 
employed  for  household  purposes.  They  consist 
of  a  pump-barrel  VV  (Fig.  162),  with  four  open- 
ings in  it,  A,  A',  B,  B'.  The  openings  A  and  B' 
are  in  communication  with  the  suction-tube  C; 
A'  and  B  are  in  communication  with  the  ejec- 
tion-tube C'.  The  four  openings  are  fitted  with 
four  valves  opening  all  in  the  same  direction, 
that  is,  from  right  to  left,  whence  it  follows 
that  A  and  B'  act  as  suction- valves,  and  A'  and 
B  as  ejection- valves,  and,  consequently,  in  whichever  direction  the 
piston  may  be  moving,  the  suction  and  ejection  of  water  are  taking 
place  at  the  same  time. 

262.  Centrifugal  Pumps.— Centrifugal  pumps,  which  have  long 
been  used  as  blowers  for  air,  and  have  recently  come  into  extensive 
use  for  purposes  of  drainage  and  irrigation,  consist  mainly  of  a  flat 
casing  or  box  of  approximately  circular  outline,  in  which  the  fluid 
is  made  to  revolve  by  a  rotating  propeller  furnished  with  fans  or 
blades.  These  extend  from  near  the  centre  outwards  to  the  circum- 
ference of  the  propeller,  and  are  usually  curved  backwards.  The 


Fig.  Ifi2. 
Double-action  Pump. 


CENTRIFUGAL  AND  JET  PUMPS. 


219 


fluid  between  them,  in  virtue  of  the  centrifugal  force  generated  by 
its  rotation,  tends  to  move  outwards,  and  is  allowed  to  pass  off 
through  a  large  conduit  which  leaves  the  case  tangentially. 


Fig.  163.-Centrifugal  Pump. 


The  first  part  of  Fig.  1G3  is  a  section  of  the  propeller  and  casing, 
C  being  "a  central  opening  at  which  the  fluid  enters,  and  D  the 
conduit  through  which  it  escapes.  The  second  part  of  the  figure 
represents  a  small  pump  as  mounted  for  use.  The  largest  class  of 


Fig.  164. — Jet  Pump. 


centrifugal  pumps  are  usually  immersed  in  the  water  to  be  pumped, 
and  revolve  horizontally. 

263.  Jet-pump. — The  jet-pump   is   a   contrivance  by  Professor 


220  PUMPS   FOR  LIQUIDS. 

James  Thomson  for  raising  water  by  means  of  the  descent  of  other 
water  from  above,  the  common  outfall  being  at  an  intermediate  level. 
Its  action  somewhat  resembles  that  of  the  blast-pipe  of  the  locomo- 
tive. The  pipe  corresponding  to  the  locomotive  chimney  must  have 
a  narrow  throat  at  the  place  where  the  jet  enters,  and  must  thence 
widen  very  gradually  towards  its  outlet,  which  is  immersed  in  the 
outfall  water  so  as  to  prevent  any  admission  of  air  during  the 
pumping.  The  water  is  drawn  up  from  the  low  level  through  a 
suction-pipe,  terminating  in  a  chamber  surrounding  the  jet-nozzle. 

Fig.  1G4  represents  the  pump  in  position,  the  jet-nozzle  with  its 
surroundings  being  also  shown  separately  on  a  larger  scale. 

The  action  of  the  jet-pump  is  explained  by  the  following  consider- 
ations. 

Suppose  we  have  a  horizontal  pipe  varying  gradually  in  sectional 
area  from  one  point  to  another,  and  completely  filled  by  a  liquid 
flowing  steadily  through  it.  Since  the  same  quantity  of  liquid  passes 
all  cross-sections  of  the  pipe,  the  velocity  will  vary  inversely  as  the 
sectional  area.  Those  portions  of  the  liquid  which  are  passing  at 
any  moment  from  the  larger  to  the  smaller  parts  of  the  pipe  are 
being  accelerated,  and  are  therefore  more  strongly  pushed  behind 
than  in  front;  while  the  opposite  is  the  case  with  those  which  are 
passing  from  smaller  to  larger.  Places  of  large  sectional  area  are 
therefore  places  of  small  velocity  and  high  pressure,  and  on  the  other 
hand,  places  of  small  area  have  high  velocity  and  low  pressure. 
Pressure,  in  such  discussions  as  this,  is  most  conveniently  expressed 
by  pressure-height,  that  is,  by  the  height  of  an  equivalent  column 
of  the  liquid.  Neglecting  friction,  it  can  be  shown  that  if  vl}  v2  be 
the  velocities  at  two  points  in  the  pipe,  and  hlt  h2  the  pressure- 
heights  at  these  points, 


g  denoting  the  intensity  of  gravity.  The  change  in  pressure-height 
is  therefore  equal  and  opposite  to  the  change  in  ^-'  Tnis  is  for  a 
horizontal  pipe. 

In  an  ascending  or  descending  pipe,  there  is  a  further  change  of 
pressure-height,  equal  and  opposite  to  the  change  of  actual  height. 

Let  H  be  the  pressure-height  at  the  free  surfaces,  that  is,  the 
height  of  a  column  of  water  which  would  balance  atmospheric  pres- 
sure; 


HYDRAULIC   PEESS. 


221 


k  the  difference  of  level  between  the  jet-nozzle  and  the  free 
surface  above  it. 

I  the  difference  of  level  between  the  jet-nozzle  and  the  free 
surface  of  the  water  which  is  to  be  raised. 

v  the  velocity  with  which  the  liquid  rushes  through  the  jet- 
nozzle, 
then    the    pressure-height    at    the   jet-nozzle    may    be    taken   as 

H  +  k  —  £-;  and  if  this  be  less  than  H  —  I  the  water  will  be  sucked 
up.     The  condition  of  working  is  therefore  that 

II  -  I  be  greater  than  H  +  Te  -  |-,  or 

v1 

£-  greater  than  k  +  I, 

where  it  will  be  observed  that  k  + 1  is  the  difference  of  levels  of  the 
highest  and  lowest  free  surfaces. 

264.  Hydraulic  Press. — The  hydraulic  press  (Fig.  1G5)  consists  of  a 
suction  and  force  pump  aa  worked  by  means  of  a  lever  turning  about 
an  axis  O.  The  water  drawn  from  the  reservoir  BB  is  forced  alonor 


Fig.  1G5.—  Bramah  Pr 


the  tube  CO  into  the  cistern  V.  In  the  top  of  the  cistern  is  an  open- 
ing through  which  moves  a  heavy  metal  plunger  AA.  This  carries 
on  its  upper  end  a  large  plate  B'B',  upon  which  are  placed  the  objects 
to  be  pressed.  Suppose  the  plunger  A  to  be  in  its  lowest  position 
when  the  pump  begins  to  work.  The  cistern  first  begins  to  fill  with 
water;  then  the  pressure  exerted  by  the  plunger  of  the  pump  is 
transmitted,  according  to  the  principles  laid  down  in  §  141,  to  the 
bottom  of  the  plunger  A;  which  accordingly  rises,  and  the  objects  to 


222 


PUMPS   FOR  LIQUIDS. 


be  pressed,  being  intercepted  between  the  plate  and  the  top  of  a  fixed 
frame,  are  subjected  to  the  transmitted  pressure.  The  amount  of 
this  pressure  depends  both  on  the  ratio  of  the  sections  of  the  pistons, 
and  on  the  length  of  the  lever  used  to  work  the  force-pump.  Sup- 
pose, for  instance,  that  the  distance  of  the  point  m,  where  the  hand 
is  applied,  from  the  point  O,  is  equal  to  twelve  times  the  distance 
10,  and  suppose  the  force  exerted  to  be  equal  to  fifty  pounds.  By 
the  principle  of  the  lever  this  is  equivalent  to  a  force  of  50  x  12  at 
the  point  I;  and  if  the  section  of  the  piston 
A  be  at  the  same  time  100  times  that  of  the 
piston  of  the  pump,  the  pressure  trans- 
mitted to  A  will  be  50  x  12  x  100  =  60,000 
pounds.  These  are  the  ordinary  conditions 
of  the  press  usually  employed  in  workshops. 
By  drawing  out  the  pin  which  serves  as  an 
axis  at  O,  and  introducing  it  at  O',  we  can 
increase  the  mechanical  advantage  of  the 
lever. 

Two  parts  essential  to  the  working  of  the 
hydraulic  press  are  not  represented  in  the 
figure.  These  are  a  safety-valve,  which 
opens  when  the -pressure  attains  the  limit 
which  is  not  to  be  exceeded;  and,  secondly, 
a  tap  in  the  tube  C,  which  is  opened  when 
we  wish  to  put  an  end  to  the  action  of  the 
press.  The  water  then  runs  off,  and  the  piston  A  descends  again  to 
the  bottom  of  the  cistern. 

The  hydraulic  press  was  clearly  described  by  Pascal,  and  at  a  still 
earlier  date  by  Stevinus,  but  for  a  long  time  remained  practically 
useless;  because  as  soon  as  the  pressure  began  to  be  at  all  strong, 
the  water  escaped  at  the  surface  of  the  piston  A.  Bramah  invented 
the  cupped  leather  cottar,  which  prevents  the  liquid  from  escaping, 
and  thus  enables  us  to  utilize  all  the  power  of  the  machine.  It  con- 
sists of  a  leather  ring  A  A  (Fig.  1GC),  bent  so  as  to  have  a  semicir- 
cular section.  This  is  fitted  into  a  hollow  in  the  interior  of  the  sides 
of  the  cistern,  so  that  water  passing  between  the  piston  and  cylinder 
will  fill  the  concavity  of  the  cupped  leather  collar,  and  by  pressing 
on  it  will  produce  a  packing  which  fits  more  tightly  as  the  pressure 
on  the  piston  increases. 

The  hydraulic  press  is  very  extensively  employed   in  the  arts. 


Fig.  ICO.-Cup-leathir. 


HYDRAULIC   PRESS.  223 

It  is  of  great  power,  and  may  "be  constructed  to  give  pressures  of 
two  or  three  hundred  tons.  It  is  the  instrument  generally  employed 
in  cases  where  very  great  force  is  required,  as  in  testing  anchors  or 
raising  very  heavy  weights.  It  was  used  for  raising  the  sections  of 
the  Britannia  tubular  bridge,  and  for  launching  the  Great  Eastern. 


CHAPTER   XXIII. 

EFFLUX  OF  LIQUIDS. — TORRICELLl'S  THEOREM. 


265.  If  an  opening  is  made  in  the  side  of  a  vessel  containing 
water,  the  liquid  escapes  with  a  velocity  which  is  greater  as  the 
surface  of  the  liquid  in  the  vessel  is  higher  above  the  orifice,  or  to 
employ  the  usual  phrase,  as  the  head  of  liquid  is  greater.  This  point 
in  the  dynamics  of  liquids  was  made  the  subject  of  experiments  by 
Torricelli,  and  the  result  arrived  at  by  him  was  that  the  velocity  of 
efflux  is  equal  to  that  which  would  be  acquired  by  a  body  falling 
freely  from  the  upper  surface  of  the  liquid  to  the  centre  of  the 
orifice.  If  h  be  this  height,  the  velocity  of  efflux  is  given  by  the 
formula 


This  is  called  Torricelli's  theorem.  It  supposes  the  orifice  to  be 
small  compared  with  the  horizontal  section  of  the  vessel,  and  to  be 
exposed  to  the  same  atmospheric  pressure  as  the  upper  surface  of  the 
liquid  in  the  vessel. 

It  may  be  deduced  from  the  principle  of  conservation  of  energy; 
for  the  escape  of  a  mass  m  of  liquid  involves  a  loss  mgh  of  energy 
of  position,  and  must  involve  an  equal  gain  of  energy  of  motion. 
But  the  gain  of  energy  of  motion  is  %mvz;  hence  we  have 

^mv1  =  mgh,  i?  =  2gh. 

The  form  of  the  issuing  jet  will  depend,  to  some  extent,  on  the 
form  of  the  orifice.  If  the  orifice  be  a  round  hole  with  sharp  edges, 
in  a  thin  plate,  the  flow  through  it  will  not  be  in  parallel  lines,  but 
the  outer  portions  will  converge  towards  the  axis,  producing  a  rapid 
narrowing  of  the  jet.  The  section  of  the  jet  at  which  this  conver- 
gence ceases  and  the  flow  becomes  sensibly  parallel,  is  called  the 
contracted  vein  or  vena  contracta.  The  pressure  within  the  jet  at 
this  part  is  atmospheric,  whereas  in  the  converging  part  it  is  greater 


CONTRACTED    VEIN. 


225 


than  atmospheric;  and  it  is  to  the  contracted  vein  that  Torricelli's 
formula  properly  applies,  v  denoting  the  velocity  at  the  contracted 
vein,  and  h  the  depth  of  its  central  point  below  the  free  surface  of 
the  liquid  in  the  vessel. 

266.  Area  of  Contracted  Vein.  Froude's  Case. — A  force  is  equal  to 
the  momentum  which  it  generates  in  the  unit  of  time.  Let  A 
denote  the  area  of  an  orifice  through  which  a  liquid  issues  horizon- 
tally, and  a  the  area  of  the  contracted  vein.  From  the  equality  of 
action  and  reaction  it  follows 'that  the  resultant  force  which  ejects 
the  issuing  stream  is  equal  and  opposite  to  the  resultant  horizontal 
force  exerted  on  the  vessel.  The  latter  may  be  taken  as  a  first 
approximation  to  be  equal  to  the  pressure  which  would  be  exerted 
on  a  plug  closing  the  orifice,  that  is  ^  f 

to  <7/iA  if  the  density  of  the  liquid  be  +^  ^* 

taken  as  unity.  k    \  /    \ 

The  horizontal  momentum  geneY- 
ated  in  the  water  in  one  second  is 
the  product  of  the  velocity  v  and  the 
mass  ejected  in  one  second.  The 
volume  ejected  in  one  second  is  va. 
This  is  equal  to  the  mass,  since  the 
density  is  unity,  and  hence  the 
momentum  is  vza,  that  is,  2gha. 
Equating  this  last  expression  for  the 
momentum  to  the  foregoing  expres- 
sion for  the  force,  we  have 


-  yhA. 


Fig.  167. 


that  is,  the  area  of  the  contracted 
vein  is  half  the  area  of  the  orifice. 

Mr.  Froude  has  pointed  out  that  this  reasoning  is  strictly  correct 
when  the  liquid  is  discharged  through  a  cylindrical  pipe  projecting 
inwards  into  the  vessel  and  terminating  with  a  sharp  edge  (Fig.  167); 
and  he  has  verified  the  result  by  accurate  experiments  in  which  the 
jet  was  discharged  vertically  downwards.  The  direction  of  flow  in 
different  parts  of  the  jet  is  approximately  indicated  by  the  arrows 
and  dotted  lines  in  the  figure;  and,  on  a  larger,  scale  by  those  in 
Fig.  1G8,  in  which  the  sections  of  the  orifice  and  of  the  contracted 
vein  are  also  indicated  by  the  lines  marked  D  and  d.  We  may 
remark  that  since  liquids  press  equally  in  all  directions,  there  can 
15 


Vr 
L 


than  on 


Fig.  103. 

their  convex 


226  EFFLUX   OF   LIQUIDS.— TOREICELLl'S   THEOREM. 

be  no  material  difference  between  the  velocities  of  a  vertical  and 
of  a  horizontal  jet  at  the  same  depth  below  the  free  surface. 

267.  Contracted  Vein  for  Orifice  in  Thin  Plate.— When  the  liquid 
is  simply  discharged  through  a  hole 
cut  in  the   side  of  the  vessel   and 
bounded  by  a  sharp  edge,  the  direc- 
tion of  flow  in  different  parts  of  the 
stream  is  shown  by  the  arrows  and 
dotted  lines  in  Fig.  169.     The  pres- 
sure on  the  sides,  in  the  neighbour- 
hood of  the  orifice,  is  less  than  that 
due  to  the  depth,  because  the  curved 
form  of  the  lines  of  flow  implies  (on 
the  principles  of   centrifugal   force) 
a  smaller  pressure  on  their  concave 
side.      The   pressure  around   the  orifice   is 
therefore  less  than  it  would  be  if  the  hole  were  plugged.     The 
unbalanced  horizontal  pressure  on  the  vessel   (if  we  suppose  the 
side  containing  the  jet  to  be  vertical)  will  therefore  exceed  the 
statical  pressure  on  the  plug  ghA,  since  the  removal  of  the  plug  not 
only  removes  the  pressure  on  the  plug  but  also  a  portion  of  the 
M         pressure  on  neighbouring  parts.     This  unbalanced  force, 
If         which  is  greater  than  ghA,  is  necessarily  equal  to  the 
]y          momentum  generated  per  second  in  the  liquid,  which  is 
still  represented  by  the  expression  v*a  or  2gha;    hence 
2gha  is   greater  than  ghA,   or  a   is    greater   than   £A. 
s'"#f...-~     Reasoning  similar  to  this  applies  to  all  ordinary  forms  of 
j(          orifice.     The  peculiarity  of  the  case  investigated  by  Mr. 
1          Froude  consists  in  the  circumstance  that  the  pressure  on 
H         the  parts  of  the  vessel  in  the  neighbourhood  of  the  orifice 
Fig.  169.    is  normai  to  the  direction  of  the  jet,  and  any  changes  in 
its  amount  which  may  be  produced  by  unplugging  the  orifice  have 
therefore  no  influence  upon  the  pressures  on  the  vessel  in  or  opposite 
to  the  direction  of  the  jet.1 

263.  Apparatus  for  Illustration. — In  the  preceding  investigations, 

1  This  section  and  the  preceding  one  are  based  on  two  communications  read  before 
the  Philosophical  Society  of  Glasgow,  February  23d  and  March  31st,  1876;  one  being 
an  extract  from  a  letter  from  Mr.  Froude  to  Sir  William  Thomson,  and  the  other  a  com- 
munication from  Professor  James  Thomson,  to  whom  we  are  indebted  for  the  accompany- 
ing illustrations. 


APPARATUS   FOR   ILLUSTRATION. 


227 


no  account  is  taken  of  friction.  When  experiments  are  conducted 
on  too  small  a  scale,  friction  may  materially  diminish  the  velocity; 
and  further,  if  the  velocity  be  tested  by  the  height  or  distance  to 
which  the  jet  will  spout,  the  resistance  of  the  air  will  diminish  this 
height  or  distance,  and  thus  make  the  velocity  appear  less  than  it 
really  is. 

Fig.  170  represents  an  apparatus  frequently  employed  for  illustrat- 


Fig.  170. — Apparatus  for  verifying  Torricelli's  Theorem. 

ing  some  of  the  consequences  of  Torricelli's  theorem.  An  upright 
cylindrical  vessel  is  pierced  on  one  side  with  a  number  of  orifices  in 
the  same  vertical  line,  which  can  be  opened  or  closed  at  pleasure. 
A  tap  placed  above  the  vessel  supplies  it  with  water,  and,  with  the 
help  of  an  overflow  pipe,  maintains  the  surface  at  a  constant  level, 
which  is  as  much  above  the  highest  orifice  as  each  orifice  is  above 
that  next  below  it.  The  liquid  which  escapes  is  received  in  a  trough, 
the  edge  of  which  is  graduated.  A  travelling  piece  with  an  index 


228  EFFLUX   OF   LIQUIDS.  —  TORRICELLl'S   THEOREM. 

line  engraved  on  it  slides  along  the  trough;  it  carries,  as  shown  in 
one  of  the  separate  figures,  a  disc  pierced  with  a  circular  hole,  and 
capable  of  being  turned  in  any  direction  about  a  horizontal  axis  pass- 
ing through  its  centre.  In  this  way  the  disc  can  always  be  placed 
in  such  a  position  that  its  plane  shall  be  at  right  angles  to  the  liquid 
jet,  and  that  the  jet  shall  pass  freely  and  exactly  through  its  centre. 
The  index  line  then  indicates  the  range  of  the  jet  with  considerable 
precision.  This  range  is  reckoned  from  the  vertical  plane  containing 
the  orifices,  and  is  measured  on  the  horizontal  plane  passing  through 
the  centre  of  the  disc.  The  distance  of  this  latter  plane  below  the 
lowest  orifice  is  equal  to  that  between  any  two  consecutive  orifices. 

The  jet,  consisting  as  it  does  of  a  series  of  projectiles  travelling  in 
the  same  path,  has  the  form  of  a  parabola. 

Let  a  be  the  range  of  the  jet,  b  the  height  of  the  orifice  above  the 
centre  of  the  ring,  and  v  the  velocity  of  discharge,  which  we  assume 
to  be  horizontal.  Then  if  t  ba  the  time  occupied  by  a  particle  of 
the  liquid  in  passing  from  the  orifice  to  the  ring,  we  have  to  express 
that  a  is  the  distance  due  to  the  horizontal  velocity  v  in  the  time  t, 
and  that  b  is  the  vertical  distance  due  to  gravity  acting  for  the  same 
time.  We  have  therefore 


But  according  to  Torricellf  s  theorem,  if  h  be  the  height  of  the  sur- 
face of  the  water  above  the  orifice,  we  have  v2  =  2gh;  and  comparing 
this  with  the  above  value  of  v-  we  deduce 


One  consequence  of  this  last  formula  is,  that  if  the  values  of  b  and 
h  be  interchanged,  the  value  of  a  will  remain  unaltered.  This 
amounts  to  saying  that  the  highest  orifice  will  give  the  same  range 
as  the  lowest,  the  highest  but  one  the  same  as  the  lowest  but  one, 
and  so  on;  a  result  which  can  be  very  accurately  verified. 

If  we  describe  a  semicircle  on  the  line  b+h,  the  length  of  an  ordi- 

3  erected  at  the  point  of  junction  of  b  and  h  is  ^,  and  since 

<*<=</  4,  bh  =2</bh,  it  follows  that  the  range  is  double  of  this  ordinate. 

This  is  on  the  hypothesis  of  no  friction.     Practically  it  is  less  than 

ible.     The  greatest  ordinate  of  the  semicircle  is  the  central  one 
and  accordingly  the  greatest  range  is  given  by  the  central  orifice. 


EFFLUX   FROM   AIR-TIGHT    SPACES. 


229 


269.  Efflux  from  Air-tight  Space. — When  the  air  at  the  free  sur- 
face of  the  liquid  in  a  vessel  is  at  a  different  pressure  from  the 
air  into  which  the  liquid  is  discharged,  we  must  express  this  differ- 
ence of  pressures  by  an  equivalent  column  of  the 
liquid,  and  the  velocity  of  efflux  will  be  that  due 
to  the  height  of  the  surface  above  the  orifice 
increased  or  diminished  by  this  column.  Efflux 
will  cease  altogether  when  the  pressure  on  the 
free  surface,  together  with  that  due  to  the  height 
of  the  free  surface  above  the  orifice,  is  equal  to 
the  pressure  outside  the  orifice;  or  if  efflux  continue 
under  such  circumstances  it  can  only  do  so  by  the 
admission  of  bubbles  of  air.  This  explains  the 
action  of  vent-pegs. 

Pipette. — This  is  a  glass  tube  (Fig.  171)  open  at 
both  ends,  and  terminating  below  in  a  small  taper- 
ing spout.  If  water  be  introduced  into  the  tube, 
either  by  aspiration  or  by  direct  immersion  in 
water,  and  if  the  upper  end  be  closed  with  the 
finger,  the  efflux  of  the  liquid  will  cease  almost 
instantly.  On  admitting  the  air  above,  the 
efflux  will  begin  again,  and  can  again  be  stopped  at  pleasure. 

The  Magic  Funnel. — This  funnel  is  double,  as  is  shown  in  Fig. 
172.  Near  the  handle  is  a 
small  opening  by  which  the 
space  between  the  two  fun- 
nels communicates  with  the 
external  air.  Another  open- 
ing connects  this  same  space 
with  the  tube  of  the  inner 
funnel.  If  the  interval  be- 
tween the  two  funnels  be 
filled  with  any  liquid,  this 
liquid  will  run  out  or  will 
cease  to  flow  according  as 
the  upper  hole  is  open  or 
closed.  The  opening  and 

closing  of  the  hole  can  be  easily  effected  with  the  thumb  of  the  hand 
holding  the  funnel  without  the  knowledge  of  the  spectator.  This 
device  has  been  known  from  very  early  times. 


Fig.  171.— Pipette. 


Fig.  172.— Magic  FunneL 


230 


EFFLUX    OF   LIQUIDS. — TOJUUCELLl's   THEOREM. 


The  instrument  may  be  used  in  a  still  more  curious  manner.  For 
this  purpose  the  space  inside  is  secretly  filled  with  highly-coloured 
wine,  which  is  prevented  from  escaping  by  closing  the  opening  above. 

Water  is  then  poured  into  the  central  funnel,  and  escapes  either 
by  itself  or  mixed  with  wine,  according  as  the  thumb  closes  or  opens 
the  orifice  for  the  admission  of  air.  In  the  second  case,  the  water 
being  coloured  with  the  wine,  it  will  appear  that  wine  alone  is 
issuing  from  the  funnel;  thus  the  operator  will  appear  to  have  the 
power  of  making  either  water  or  wine  flow  from  the  vessel  at  his 
pleasure. 

The  Inexhaustible  Bottle.— The  inexhaustible  bottle  (Fig.  173)  is 
a  toy  of  the  same  kind.  It  is  an  opaque  bottle  of  sheet-iron  or 


Fig.  173.— Inexhaustible  Bottle. 


gutta-percha,  containing  within  it  five  small  vials.  These  communi- 
cate with  the  exterior  by  five  small  holes,  which  can  be  closed  by  the 
five  fingers  of  the  hand.  Each  vial  has  also  a  small  neck  which 
passes  up  the  large  neck  of  the  bottle.  The  five  vials  are  filled  with 
five  different  liquids,  any  one  of  which  can  be  poured  out  at  pleasure 
by  uncovering  the  corresponding  hole. 

270.  Intermittent  Fountain.-The  intermittent  fountain  is  an 
apparatus  analogous  to  the  preceding,  except  that  the  interruptions 
m  the  efflux  are  produced  automatically  by  the  action  of  the  instru- 


INTERMITTENT   FOUNTAIN. 


231 


ment,  without  the  intervention  of  the  operator.  It  consists  of  a  globe 
V  (Fig.  174),  which  can  be  closed  air-tight  by  means  of  a  stopper,  and 
is  in  communication  with  efflux  tubes  a,  which  discharge  into  a  basin 
B,  having  a  small  hole  o  in  its  bottom  for  permitting  the  water  to 
escape  into  a  lower  basin  C. 
A  central  tube  t,  open  at  both 
ends,  extends  nearly  to  the  top 
of  the  globe,  and  nearly  to  the 
bottom  of  the  basin  B. 

Suppose  the  globe  to  be  filled 
with  water,  the  basins  being 
empty.  Then  the  water  will 
flow  from  the  efflux  tubes  a, 
while  air  will  pass  up  through 
the  central  tube.  As  the  water 
issues  from  the  efflux  tubes 
much  faster  than  it  escapes 
through  the  opening  o,  the  level 
rises  in  the  basin  B  till  the 
lower  end  of  the  tube  t  is 
covered.  The  pressure  of  the 
air  in  the  upper  part  of  the 
globe  then  rapidly  diminishes, 
and  the  efflux  from  the  tubes 
a  is  stopped.  But  as  the  water 
continues  to  escape  from  the 
basin  B  through  the  opening  o,  the  bottom  of  the  tube  t  is  again 
uncovered,  the  liquid  again  issues  from  the  efflux  tubes,  and  the 
same  changes  are  repeated. 

271.  Siphon. — The  siphon  is  an  instrument  in  which  a  liquid, 
under  the  combined  action  of  its  own  weight  and  atmospheric  pres- 
sure, flows  first  up-hill  and  then  down-hill,  but  always  in  such  a  way 
as  to  bring  about  a  lowering  of  the  centre  of  gravity  of  the  whole 
liquid  mass. 

In  its  simplest  form,  it  consists  of  a  bent  tube,  one  end  of  which 
is  immersed  in  the  liquid  to  be  removed,  while  the  other  end  either 
discharges  into  the  air,  at  a  lower  level  than  the  surface  of  the  liquid 
in  the  vessel,  as  in  Fig.  175,  or  dips  into  the  liquid  of  a  receiving 
vessel,  the  surface  of  this  liquid  being  lower  than  that  of  the  liquid 
in  the  discharging  vessel. 


Fig  174.  — Intermittent  Fountain. 


232  EFFLUX   OF   LIQUIDS.— TORRICELLIS   THEOREM. 

We  shall  discuss  the  latter  case,  and  shall  denote  the  difference  of 
levels  of  the  two  surfaces  by  h,  while  the  height  of  a  column  of  the 
liquid  equivalent  to  atmospheric  pressure  will  be  denoted  by  H. 

Let  the  siphon  be  full  of  liquid,  and  imagine  a  diaphragm  to  be 
drawn  across  it  at  any  point,  so  as  to  prevent  flow.  Let  this  dia- 


Fig.  175.— Siphor 


phragm  be  at  a  height  x  above  the  higher  of  the  two  free  surfaces, 
and  at  a  height  y  above  the  lower,  so  that  we  have 

y  -  x  =  h. 

The  pressure  on  the  side  of  the  diaphragm  next  the  higher  free  sur- 
face will  be  H  —  x,  (pressure  being  expressed  in  terms  of  the  equiva- 
lent liquid  column,)  and  the  pressure  on  the  other  side  of  the  dia- 
phragm will  be  H  —  y,  which  is  less  than  the  former  by  y  —  x,  that 
is  by  h.  The  diaphragm  therefore  experiences  a  resultant  force  due 
to  a  depth  h  of  the  liquid,  urging  it  from  the  higher  to  the  lower  free 
surface,  and  if  the  diaphragm  be  removed,  the  liquid  will  be  pro- 
pelled in  this  direction. 

In  practice,  the  two  legs  of  the  siphon  are  usually  of  unequal 
length,  and  the  flow  is  from  the  shorter  to  the  longer ;  but  this  is  by 
no  means  essential,  for  by  a  sufficiently  deep  immersion  of  the  long 


SIPHON. 


233 


leg,  the  direction  of  flow  may  be  reversed.  The  direction  of  flow 
depends  not  on  the  lengths  of  the  legs,  but  on  the  levels  of  the  two 
free  surfaces. 

If  the  liquid  in  the  discharging  vessel  falls  below  the  end  of  the 
siphon,  or  if  the  siphon  is  lifted  out  of  it,  air  enters,  and  the  siphon 
is  immediately  emptied  of  liquid.  If  the  liquid  in  the  receiving 
vessel  is  removed,  so  that  the  discharging  end  of  the  siphon  is  sur- 
rounded by  air,  as  in  the  figure,  the  flow  will  continue,  unless  air 
bubbles  up  the  tube  and  breaks  the  liquid  column.  This  interrup- 
tion is  especially  liable  to  occur  in  large  tubes.  It  can  be  prevented 
by  bending  the  end  of  the  siphon  round,  so  as  to  discharge  the 
liquid  in  an  ascending  direction.  To  adapt  the  foregoing  investi- 
gation to  the  case  of  a  siphon  discharging  into  air,  we  have  only  to 
substitute  the  level  of  the  discharging  end  for  the  level  of  the  lower 
free  surface,  so  that  y  will  denote  the  depth  of  the  discharging  end 
below  the  diaphragm,  and  h  its  depth  below  the  surface  of  the  liquid 
which  is  to  be  drawn  off. 

As  the  ascent  of  the  liquid  in  the  siphon  is  due  to  atmospheric 
pressure  on  the  upper  free  sur- 
face, it  is  necessary  that  the 
highest  point  of  the  siphon  (if 
intended  for  water)  should  not 
be  more  than  about  33  feet 
above  this  surface. 

272.  Starting  the  Siphon.— In 
order  to  make  a  siphon  begin 
working,  we  must  employ  means 
to  fill  it  with  the  liquid.  This 
can  sometimes  be  done  by  dip- 
ping it  in  the  liquid,  and  then 
placing  it  in  position  while  the 
ends  are  kept  closed;  or  by  in- 
serting one  end  in  the  liquid 
which  we  wish  to  remove,  and 
sucking  at  the  other.  It  is  usu- 
ally more  convenient  to  apply  suction  by  means  of  a  side  tube, 
as  in  Fig.  176,  this  tube  being  sometimes  provided  with  an 
enlargement  to  prevent  the  liquid  from  entering  the  mouth.  One 
end  of  the  siphon  is  inserted  in  the  liquid  which  is  to  be  removed, 
while  the  other  end  is  stopped,  and  the  operator  applies  suction  at 


Fig.  ITG.-SUrting  the  Siplion. 


234  EFFLUX   OF   LIQUIDS.— TORRICELLI'S   THEOREM. 

the  side  tube  till  the  liquid  flows  over.  In  siphons  for  commercial 
purposes,  the  suction  is  usually  produced  by  a  pump. 
f  273.  Siphon  for  Sulphuric  Acid.— Fig.  177  represents  a  siphon  used 
for  transferring  sulphuric  acid  from  one  vessel  to  another.  The 
long  branch  is  first  filled  with  sulphuric  acid.  This  is  effected  by 
means  of  two  funnels  (which  can  be  plugged  at  pleasure)  at  the 
bend  of  the  tube.  One  of  these  admits  the  liquid,  and  the  other 
suffers  the  air  to  escape.  The  two  funnels  are  then  closed,  and 

the  tap  at  the  lower  end 
of  the  tube  is  opened  so  as  to 
allow  the  liquid  to  escape. 
The  air  in  the  short  branch 
follows  the  acid,  and  becomes 
rarefied;  the  acid  behind  it 
rises,  and  if  it  passes  the  bend, 
the  siphon  will  be  started,  for 
each  portion  of  the  liquid 
which  issues  from  the  tube  will 
draw  an  equal  portion  from 

Fig.  177.-Siphon  for  Sulphuric  Acid.  ^  ^^  ^Q  ^  jong  kranch 

To. insure  the  working  of  the  sulphuric  acid  siphon,  it  is  not  suffi- 
cient to  have  the  vertical  height  of  the  long  branch  greater  than  that 
of  the  short  branch;  it  is  farther  necessary  that  it  should  exceed  a 
certain  limit,  which  depends  upon  the  dimensions  of  the  siphon  in 
each  particular  case.  In  order  to  calculate  this  limit,  we  must 
remark  that  when  the  liquid  begins  to  flow,  its  height  diminishes  in 
the  long  and  increases  in  the  short  branch;  if  these  two  heights 
should  become  equal,  there  would  be  equilibrium.  We  see,  then, 
that  in  order  that  the  siphon  may  work,  it  is  necessary  that  when 
the  liquid  rises  to  the  bend  of  the  tube,  there  should  be  in  the  long 
branch  a  column  of  liquid  whose  vertical  height  is  at  least  equal  to 
that  of  the  short  branch,  which  we  shall  denote  by  h,  and  the  actual 
length  of  the  short  branch  from  the  surface  of  the  liquid  in  which 
it  dips  to  the  summit  of  the  bend  by  h'.  Then  if  a  be  the  inclina- 
tion of  the  long  branch  to  the  vertical,  and  L  the  length  of  the  long 
branch,  which  we  suppose  barely  sufficient,  the  length  of  the  column 
of  liquid  remaining  in  the  long  branch  will  be  h  sec  a.  The  air 
which  at  atmospheric  pressure  H  occupied  the  length  h',  now  under 
the  pressure  H  —  h  occupies  a  length  L  —  h  sec  a;  hence  by  Boyle's 
law,  we  have 


CUP   OF  TANTALUS. 


235 


Ilk'  =  (H  -  h)  (L-h  sec  a),  whence  L  =  h  sec  a  + 


Fig.  ITS.— Vase  of  Tantalus. 


In  this  formula  H  denotes  the  height  of  a  column  of  sulphuric  acid 
whose  pressure  equals  that  of  the  atmosphere. 

•>  274.  Cup  of  Tantalus.—  The  siphon  may  be  employed  to  produce 
the  intermittent  flow  of  a  liquid.  Suppose,  for  instance,  that  we 
have  a  cup  (Fig.  178)  in  which  is 
a  bent  tube  rising  to  a  height  n, 
and  with  the  short  branch  termi- 
nating near  the  bottom  of  the 
cup,  while  the  long  branch  passes 
through  the  bottom.  If  liquid  be 
poured  into  the  cup,  the  level  will 
gradually  rise  in  the  short  branch 
of  the  bent  tube,  till  it  reaches 
the  summit  of  the  bend,  when  the 
siphon  will  begin  to  discharge  the 
liquid.  If  the  liquid  then  escapes 
by  the  siphon  faster  than  it  is 

poured  into  the  vessel,  the  level  of  the  liquid  in  the  cup  will  gradu- 
ally fall  below  the  termination  of  the  shorter  branch.  The  siphon 
will  then  empty  itself,  and  will  not  recommence  its  action  till  the 
liquid  has  again  risen  to  the  level  of  the  bend. 

The  siphon  may  be  concealed  in  the  interior  of  the  figure  of  a 
man  whose  mouth  is  just  above  the  top  of  the  siphon.  If  water  be 
poured  in  very  slowly,  it  will  continually  rise  nearly  to  his  lips  and 
then  descend  again.  Hence  the  name.  Instead  of  a  bent  tube  we 
may  employ,  as  in  the  first  figure,  a  straight  tube  covered  by  a  bell- 
glass  left  open  below;  in  this  case  the  space  between  the  tube  and 
the  bell  takes  the  place  of  the  shorter  leg  of  the  siphon. 

It  is  to  an  action  of  this  kind  that  natural  intermittent  springs  are 
generally  attributed.  Suppose  a  reservoir  (Fig.  179)  to  communicate 
with  an  outlet  by  a  bent  tube  forming  a  siphon,  and  suppose  it  to 
be  fed  by  a  stream  of  water  at  a  slower  rate  than  the  siphon  is  able 
to  discharge  it.  When  the  water  has  reached  the  bend,  the  siphon 
will  become  charged,  and  the  reservoir  will  be  emptied;  flow  will 
then  cease  until  it  becomes  charged  again. 

-  275.  Mariotte's  Bottle.  —  This  is  an  apparatus  often  employed  to  ob- 
tain a  uniform  flow  of  water.  Through  the  cork  at  the  top  of  the 
bottle  (Fig.  180)  passes  a  straight  vertical  tube  open  at  both  ends,  and 


236 


EFFLUX   OF  LIQUIDS.— TORRICELLI'S   THEOREM. 


in  one  side  of  the  bottle  near  the  bottom  is  a  second  opening  furnished 
with  a  horizontal  efflux  tube  b  at  a  lower  level  than  the  lower 
end  of  the  vertical  tube.  Suppose  that  both  the  bottle  and  the 
vertical  tube  are  in  the  first  instance  full  of  water,  and  that  the 
efflux  tube  is  then  opened.  The  liquid  flows  out,  and  the  vertical 
tube  is  rapidly  emptied.  Air  then  enters  the  bottle  through  the 
vertical  tube,  and  bubbles  up  from  its  lower  end  a  through  the 
liquid  to  the  upper  part  of  the  bottle.  As  soon  as  this  process 
begins,  the  velocity  of  efflux,  which  up  to  this  point  has  been 
rapidly  diminishing  (as  is  shown  by  the  diminished  range  of  the 


Fig.  179. -Intermittent  Spring. 

jet),  becomes  constant,  and  continues  so  till  the  level  of  the  liquid 
has  fallen  to  a,  after  which  it  again  diminishes.  During  the  time 
of  constant  flow,  the  velocity  of  efflux  is  that  due  to  the  height  of 
a  above  6,  and  the  air  in  the  upper  part  of  the  bottle  is  at  less  than 
atmospheric  pressure,  the  difference  being  measured  by  the  height 
of  the  surface  of  the  liquid  above  a.  Strictly  speaking,  since  the 
air  enters  not  in  a  continuous  stream  but  in  bubbles,  there  must  be 
slight  oscillations  of  velocity,  keeping  time  with  the  bubbles,  but 
they  are  scarcely  perceptible. 

Instead  of  the  vertical  tube,  we  may  have  a  second  opening  in  the 


MARIOTTE  S   BOTTLE. 


237 


side  of  the  bottle,  at  a  higher  level  than  the  first;  as  shown  in  Fig. 
180.  Air  will  enter  through  the  pipe  a,  which  is  fitted  in  this  upper 
opening,  and  the  liquid  will  issue  at  the  lower  pipe  6,  with  a  constant 
velocity  due  to  the  height  of  a  above  b. 

Mariotte's  bottle  is  sometimes  used  in  the  laboratory  to  produce 


Fig.  ISO.— Mariotte's  Bottle. 

the  uniform  flow  of  a  gas  by  employing  the  water  which  escapes  to 
expel  the  gas.  We  may  also  draw  in  gas  through  the  tube  of 
Mariotte's  bottle;  in  this  case,  the  flow  of  the  water  is  uniform,  but 
the  flow  of  the  gas  is  continually  accelerated,  since  the  space  occupied 
by  it  in  the  bottle  increases  uniformly,  but  the  density  of  the  gas  in 
this  space  continually  increases. 


EXAMPLES. 


PARALLELOGRAM  OF  VELOCITIES,  AND  PARALLELOGRAM  OF  FORCES. 

I.  A  ship  sails  through  the  water  at  the  rate  of  10  miles  per  hour,  and  a  ball 
rolls  across  the  deck  in  a  direction  perpendicular  to  the  course,  at  the  same  rate. 
Find  the  velocity  of  the  ball  relative  to  the  water. 

,  2.  The  wind  blows  from  a  point  intermediate  between  N.  and  E.  The  nor- 
thei-ly  component  of  its  velocity  is  5  miles  per  hour,  and  the  easterly  component 
is  12  miles  per  hour.  Find  the  total  velocity. 

3.  The  wind  is  blowing  due  N.E.  with  a  velocity  of  10  miles  an  hour.     Find 
the  northerly  and  easterly  components. 

4.  Two  forces  of  6  and  8  units  act  upon  a  body  in  lines  which  meet  in  a  point 
and  are  at  right  angles.     Find  the  magnitude  of  their  resultant. 

5.  Two  equal  forces  of  100  units  act  upon  a  body  in  lines  which  meet  in  a 
point  and  are  at  right  angles.     Find  the  magnitude  of  their  resultant. 

6.  A  force  of  100  units  acts  at  an  inclination  of  45°  to  the  horizon.     Resolve 
it  into  a  horizontal  and  a  vertical  component. 

7.  Two  equal  forces  act  in  lines  which  meet  in  a  point,  and  the  angle  between 
their  directions  is  120°.     Show  that  the  resultant  is  equal  to  either  of  the  forces. 

8.  A  body  is  pulled  north,  south,  east,  and  west  by  four  strings  whose  direc- 
tions meet  in  a  point,  and  the  forces  of  tension  in  the  strings  are  equal  to  10,  15, 
20,  and  32  Ibs.  weight  respectively.     Show  that  the  resultant  is  equal  to  13  Ibs. 
weight. 

9.  Five  equal  forces  act  at  a  point,  in  one  place.     The  angles  betwteii  the  first 
and  second,  between  the  second  and  third,  between  the  third  and  fourth,  and 
between  the  fourth  and  fifth,  are  each  60°.     Find  their  resultant. 

10.  If  6  be  the  angle  between  the  directions  of  two  forces  P  and  Q  acting  at  a 
point,  and  R  be  their  resultant,  show  that 

E2  =  Ps  +  Q2  +  2PQ  cos  0. 

II.  Show  that  the  resultant  of  two  equal  forces  P,  acting  at  an  angle  6,  is 
2P  cos  4J. 

PARALLEL  FORCES,  AND  CENTRE  OF  GRAVITY. 

10*.  A  straight  rod  10  ft.  long  is  supported  at  a  point  3  ft.  from  one  end. 
What  weight  hung  from  this  end  will  be  supported  by  12  Ibs.  hung  from  the 
other,  the  weight  of  the  rod  being  neglected  ? 

11*.  Weights  of  15  and  20  Ibs.  are  hung  from  the  two  ends  of  a  straight  rod 
70  in.  long.  Find  the  point  about  which  the  rod  will  balance,  its  own  weight 
being  neglected. 


240  EXAMPLES. 

12.  A  weight  of  100  Ibs.  is  slung  from  a  pole  which  rests  on  the  shoulders  of 
two  men,  A  and  B.     The  distance  between  the  points  where  the  pole  presses  their 
shoulders  is  10  ft.,  and  the  point  where  the  weight  is  slung  is  4  ft.  from  the  point 
where  the  pole  presses  on  A's  shoulder.     Find  the  weight  borne  by  each,  the 
weight  of  the  pole  being  neglected. 

13.  A  uniform  straight  lever  10  ft.  long  balances  at  a  point  3  ft.  from  one  end, 
when  12  Ibs.  are  hung  from  this  end  and  an  unknown  weight  from  the  other. 
The  lever  itself  weighs  8  Ibs.     Find  the  unknown  weight. 

14.  A  straight  lever  6  ft.  long  weighs  10  Ibs.,  and  its  centre  of  gravity  is  4  ft. 
from  one  end.     What  weight  at  this  end  will  support  20  Ibs.  at  the  other,  when 
the  lever  is  supported  at  1  ft.  distance  from  the  latter? 

15.  Two  equal  weights  of  10  Ibs.  each  are  hung  one  at  each  end  of  a  straight 
lever  6  ft.  long,  which  weighs  5  Ibs.;  and  the  lever,  thus  weighted,  balances  about 
a  point  3  in.  distant  from  the  centre  of  its  length.     Find  its  centre  of  gravity. 

16.  A  uniform  lever  10  ft.  long  balances  about  a  point  1  ft.  from  one  end, 
when  loaded  at  that  end  with  50  Ibs.     Find  the  weight  of  the  lever. 

17.  A  straight  lever  10  ft.  long,  when  unweighted,  balances  about  a  point  4  ft. 
from  one  end ;  but  when  loaded  with  20  Ibs.  at  this  end  and  4  Ibs.  at  the  other, 
it  balances  about  a  point  3  ft.  from  the  end.     Find  the  weight  of  the  lever. 

18.  A  lever  is  to  be  cut  from  a  bar  weighing  3  Ibs.  per  ft     What  must  be  its 
length  that  it  may  balance  about  a  point  2  ft.  from  one  end,  when  weighted  at 
this  end  with  50  Ibs.]     (The  solution  of  this  question  involves  a  quadratic  equa- 
tion.) 

19.  A  lever  is  supported  at  its  centre  of  gravity,  which  is  nearer  to  one  end 
than  to  the  other.     A  weight  P  at  the  shorter  arm  is  balanced  by  2  Ibs.  at  the 
longer;  and  the  same  weight  P  at  the  longer  arm  is  balanced  by  18  Ibs.  at  the 
shorter.     Find  P. 

20.  Weights  of  2,  3,  4  and  5  Ibs.  are  hung  at  points  distant  respectively  1,  2, 
3  and  4  ft.  from  one  end  of  a  lever  whose  weight  may  be  neglected.     Find  the 
point  about  which  the  lever  thus  weighted  will  balance.     (This  and  the  following 
questions  are  best  solved  by  taking  moments  round  the  end  of  the  lever.     The 
sum  of  the  moments  of  the  four  weights  is  equal  to  the  moment  of  their  resul- 
tant.) 

21.  Solve  the  preceding  question,  supposing  the  lever  to  be  5  ft.  long,  uniform, 
and  weighing  2  Ibs. 

22.  Find,  in  position  and  magnitude,  the  resultant  of  two  parallel  and  oppo- 
sitely directed  forces  of  10  and  12  units,  their  lines  of  action  being  1  yard  apart. 

23.  A  straight  lever  without  weight  is  acted  on  by  four  parallel  forces  at  the 
following  distances  from  one  end : — 

At  1  ft.,  a  force  of  2  units,  acting  upwards. 
At  2  ft.,         „          3     „          „       downwards. 
At  3  ft.,         „          4     „          „       upwards. 
At  4  ft.,         „          5     „          „       downwards. 

Where  must  the  fulcrum  be  placed  that  the  lever  may  be  in  equilibrium,  and 
what  will  be  the  pressure  against  the  fulcrum? 

24.  A  straight  lever,  turning  freely  about  an  axis  at  one  end,  is  acted  on  by 
four  parallel  forces,  namely— 


EXAMPLES.  241 

A  downward  force  of  3  Ibs.  at  1  ft.  from  axis. 
A  downward  force  of  5      „       3  ft.          „ 
An  upward  force  of     4      „       2  ft.          „ 
An  upward  force  of     ti      „       4  ft.          „ 

What  must  be  the  weight  of  the  lever  that  it  may  be  in  equilibrium,  its  centre  of 
gravity  being  3  ft.  from  the  axis? 

25.  In  a  pair  of  nut-crackers,  the  nut  is  placed  one  inch  from  the  hinge,  and 
the  hand  is  applied  at  a  distance  of  six  inches  from  the  hinge.     How  much 
pressure  must  be  applied  by  the  hand,  if  the  nut  requires  a  pressure  of  13  Ibs.  to 
break  it,  and  what  will  be  the  amount  of  the  pressure  on  the  hinges? 

26.  In  the  steelyard,  if  the  horizontal  distance  between  the  fulcrum  and  the 
knife-edge  which  supports  the  body  weighed  be  3  in.,  and  the  movable  weight  be 
7  Ibs.,  how  far  must  the  latter  be  shifted  for  a  difference  of  1  Ib.  in  the  body 
weighed  ? 

27.  The  head  of  a  hammer  weighs  20  Ibs.  and  the  handle  2  Ibs.     The  distance 
between  their  respective  centres  of  gravity  is  24  inches.     Find  the  distance  of  the 
centre  of  gravity  of  the  hammer  from  that  of  the  head. 

28.  One  of  the  four  triangles  into  which  a  square  is  divided  by  its  diagonals  is 
removed.     Find  the  distance  of  the  centre  of  gravity  of  the  remainder  from  the 
intersection  of  the  diagonals. 

29.  A  square  is  divided  into  four  equal  squares  and  one  of  these  is  removed. 
Find  the  distance  of  the  centre  of  gravity  of  the  remaining  portion  from  the 
centre  of  the  original  square. 

30.  Find  the  centre  of  gravity  of  a  sphere  1  decimetre  in  radius,  having  in  its 
interior  a  spherical  excavation  whose  centre  is  at  a  distance  of  5  centimetres  from 
the  centre  of  the  large  sphere  and  whose  radius  is  4  centimetres. 

31.  Weights  P,  Q,  R,  S  are  hung  from  the  corners  A,  B,  C,  D  of  a  uniform 
square  plate  whose  weight  is  W.     Find  the  distances  from  the  sides  AB,  AD  of 
the  point  about  which  the  plate  will  balance. 

32.  An  isosceles  triangle  stands  upon  one  side  of  a  square  as  base,  the  altitude 
of  the  triangle  being  equal  to  a  side  of  llie  square.     Show  that  the  distance  of  the 
centre  of  the  whole  figure  from  the  opposite  side  of  the  square  is  |-  of  a  side  of  the 
square. 

33.  A  right  cone  stands  upon  one  end  of  a  right  cylinder  as  base,  the  altitude 
of  the  cone  being  equal  to  the  height  of  the  cylinder.     Show  that  the  distance  of 
the  centre  of  the  whole  volume  from  the  opposite  end  of  the  cylinder  is  |J  of  the 
height  of  the  cylinder. 

WORK  AND  STABILITY. 

34.  A  body  consists  of  three  pieces,  whose  masses  are  as  the  numbers  1,  3,  9; 
aud  the  centres  of  these  masses  are  at  heights  of  2,  3,  and  5  cm.  above  a  certain 
level.     Find  the  height  of  the  centre  of  the  whole  mass  above  this  level. 

35.  The  body  above-mentioned  is  moved  into  a  new  position,  in  which  the 
heights  of  the  centres  of  the  three  masses  are  1,  3,  and  7  cm.     Find  the  new 
height  of  the  centre  of  the  whole  mass. 

36.  Find  the  work  done  against  gravity  in  moving  the  body  from  the  first 
position  into  the  second ;  employing  as  the  unit  of  work  the  work  done  in  raising 
the  smallest  of  the  three  pieces  through  1  cm. 

16 


242  EXAMPLES. 

37  Find  the  portions  of  this  work  done  in  moving  each  of  the  three  pieces. 

38  The  dimensions  of  a  rectangular  block  of  stone  of  weight  W  are  AB  =  a 
AC  =  b,  AD  =  c,  and  the  edges  AB,  AC  are  initially  horizontal.     How  much 
work  is  done  against  gravity  in  tilting  the  stone  round  the  edge  AB 

^  39*1  chain  of  weight  W  and  length  I  hangs  freely  by  its  upper  end  which  is 
attached  to  a  drum  upon  which  the  chain  can  be  wound,  the  diameter  of  the  drum 
being  small  compared  with  I.  Compute  the  work  done  against  gravity  in  winding 
up  two-thirds  of  the  chain. 

40  Two  equal  and  similar  cylindrical  vessels  with  their  bases  at  the  same 
level  contain  water  to  the  respective  heights  h  and  H  centimetres,  the  area  of 
either  base  being  a  sq.  cm.  Find,  in  gramme-centimetres,  the  work  done  by 
gravity  in  equalizing' the  levels  when  the  two  vessels  are  connected. 
"  41.  Two  forces  acting  at  the  ends  of  a  rigid  rod  without  weight  equilibrate 
each  other.  Show  that  the  equilibrium  is  stable  if  the  forces  are  pulling  outwards 
and  unstable  if  they  are  pushing  inwards. 

42.  Two  equal  weights  hanging  from  the  two  ends  of  a  string,  which  passes 
over  a  fixed  pulley  without  friction,  balance  one  another.     Show  that  the  equili- 
brium is  neutral  if  the  string  is  without  weight,  and  is  unstable  if  the  string  is 
heavy. 

43.  Show  that  a  uniform  hemisphere  resting  on  a  horizontal  plane  has  two 
positions  of  stable  equilibrium.    Has  it  any  positions  of  unstable  equilibrium  1 

INCLINED  PLANE,  &c. 

44.  On  an  inclined  plane  whose  height  is  J  of  its  length,  what  power  acting 
parallel  to  the  plane  will  sustain  a  weight  of  112  Ibs.  resting  on  the  plane  without 
friction? 

45.  The  height,  base,  and  length  of  an  inclined  plane  are  as  the  numbers  3, 
4,  5.     What  weight  will  be  sustained  on  the  plane  without  friction  by  a  power  of 
100  Ibs.  acting  (a)  parallel  to  the  base,  (6)  parallel  to  the  plane  ? 

46.  Find  the  ratio  of  the  power  applied  to  the  pressure  produced  in  a  screw- 
press  without  friction,  the  power  being  applied  at  the  distance  of  1  ft.  from  the 
axis  of  the  screw,  and  the  distance  between  the  threads  being  ^  in. 

47.  In  the  system  of  pulleys  in  which  one  cord  passes  round  all  the  pulleys, 
its  different  portions  being  parallel,  what  power  will  sustain  a  weight  of  2240  Ibs. 
without  friction,  if  the  number  of  cords  at  the  lower  block  be  61? 

48.  A  balance  has  unequal  arms,  but  the  beam  assumes  the  horizontal  position 
when  both  scale-pans  are  empty.     Show  that  if  the  two  apparent  weights  of  a 
body  are  observed  when  it  is  placed  first  in  one  pan  and  then  in  the  other,  the 
true  weight  will  be  found  by  multiplying  these  together  and  taking  the  square 
root. 

FORCE,  MASS,  AND  VELOCITY. 

The  motion  is  supposed  to  le  rectilinear. 

49.  A  force  of  1000  dynes  acting  on  a  certain  mass  for  one  second  gives  it  a 
velocity  of  20  cm.  per  sec.     Find  the  mass  in  grammes. 

50.  A  constant  force  acting  on  a  mass  of  12  gm.  for  one  sec.  gives  it  a  velocity 
of  6  cm.  per  sec.     Find  the  force  in  dynes. 


EXAMPLES.  243 

51.  A  force  of  490  dynes  acts  on  a  mass  of  70  gin.  for  one  sec.     Find  the 
velocity  generated. 

52.  In  the  preceding  example,  if  the  time  of  action  be  increased  to  5  sec.,  what 
will  be  the  velocity  generated? 

In  the  following  examples  the  unit  of  momentum  referred  to  is  the  momentum  of  a  gramme  moving 
with  a,  velocity  of  a,  centimetre  per  second. 

53.  What  is  the  momentum  of  a  mass  of  15  gm.  moving  with  a  velocity  of 
translation  of  4  cm.  per  sec.? 

54.  What  force,  acting  upon  the  mass  for  1  sec.,  would  produce  this  velocity? 

55.  What  force,  acting  upon  the  mass  for  10  sec.,  would  produce  the  same 
velocity? 

56.  Find  the  force  which,  acting  on  an  unknown  mass  for  12  sec.,  would  pro- 
duce a  momentum  of  84. 

57.  Two  bodies  initially  at  rest  move  towards  each  other  in  obedience  to 
mutual  attraction.     Their  masses  are  respectively  1  gm.  and  100  gm.     If  the  force 
of  attraction  be  TJ^  of  a  dyne,  find  the  velocity  acquired  by  each  mass  in  1  sec. 

58.  A  gun  is  suspended  by  strings  so  that  it  can  swing  freely.     Compare  the 
velocity  of  discharge  of  the  bullet  with  the  velocity  of  recoil  of  the  gun;   the 
masses  of  the  gun  and  bullet  being  given,  and  the  mass  of  the  powder  being 
neglected. 

59.  A  bullet  fired  vertically  upwards,  enters  and  becomes  imbedded  in  a  block 
of  wood  falling  vertically  overhead ;  and  the  block  is  brought  to  rest  by  the  im- 
pact.    If  the  velocities  of  the  bullet  and  block  immediately  before  collision  were 
respectively  1500  and  100  ft.  per  sec.,  compare  their  masses. 

FALLING  BODIES  AND  PROJECTILES. 

Assuming  that  a  falling  body  acquires  a  velocity  of  980  cm.  per  sec.  by  falling 
for  1  sec.,  find : — 

60.  The  velocity  acquired  in  ^  of  a  second. 

61.  The  distance  passed  over  in  ^  sec. 

62.  The  distance  that  a  body  must  fall  to  acquire  a  velocity  of  980  cm.  per  sec. 

63.  The  time  of  rising  to  the  highest  point,  when  a  body  is  thrown  vertically 
upwards  with  a  velocity  of  6860  cm.  per  sec. 

64.  The  height  to  which  a  body  will  rise,  if  thrown  vertically  upwards  with  a 
velocity  of  490  cm.  per  sec. 

65.  The  velocity  with  which  a  body  must  be  thrown  vertically  upwards  that 
it  may  rise  to  a  height  of  200  cm. 

66.  The  velocity  that  a  body  will  have  after  ^  sec.,  if  thrown  vertically  up- 
wards with  a  velocity  of  300  cm.  per  sec. 

67.  The  point  that  the  body  in  last  question  will  have  attained. 

68.  The  velocity  that  a  body  will  have  after  2£  sees.,  if  thrown  vertically  up- 
wards with  a  velocity  of  800  crn.  per  sec. 

69.  The  point  that  the  body  in  last  question  will  have  reached. 
Assuming  that  a  falling  body  acquires  a  velocity  of  32  ft.  per  sec.  by  falling  for 

1  sec.,  find : — 

70.  The  velocity  acquired  in  12  sec. 

71.  The  distance  fallen  in  12  sec. 


244  EXAMPLES. 

72.  The  distance  that  a  body  must  fall  to  acquire  a  velocity  of  10  ft.  per  sec. 

73.  The  time  of  rising  to  the  highest  point,  when  a  body  is  thrown  vertically 
upwards  with  a  velocity  of  160  ft.  per  sec. 

74.  The  height  to  which  a  body  will  rise,  if  thrown  vertically  upwards  with  a 
velocity  of  32  ft.  per  sec. 

75.  The  velocity  with  which  a  body  must  be  thrown  vertically  upwards  that 
it  may  rise  to  a  height  of  25  ft. 

76.  The  velocity  that  a  body  will  have  after  3  sec.,  if  thrown  vertically  up- 
wards with  a  velocity  of  100  ft.  per  sec. 

77.  The  height  that  the  body  in  last  question  will  have  ascended. 

78.  The  velocity  that  a  body  will  have  after  1^  sec.,  if  thrown  vertically  down- 
wards with  a  velocity  of  30  ft.  per  sec. 

79.  The  distance  that  the  body  in  last  question  will  have  described. 

80.  A  body  is  thrown  horizcntally  from  the  top  of  a  tower  100  m.  high  with 
a  velocity  of  30  metres  per  sec.     When  and  where  will  it  strike  the  ground? 

81.  Two  bodies  are  successively  dropped  from  the  same  point,  with  an  interval 
of  \  of  a  second.     When  will  the  distance  between  them  be  one  metre? 

82.  Show  that  if  x  and  y  are  the  horizontal  and  vertical  co-ordinates  of  a  pro- 
jectile referred  to  the  point  of  projection  as  origin,  their  values  after  time  t  are 

x  =  Vt  cos  a,  y  =  "Vt  sin  a  -  £  gt*. 

83.  Show  that  the  equation  to  the  trajectory  is 

y  =  *tana-^k' 

and  that  if  V  and  a,  can  be  varied  at  pleasure,  the  projectile  can  in  general  be 
made  to  traverse  any  two  given  points  in  the  same  vertical  plane  with  the  point 
of  projection. 

ATWOOD'S  MACHINE. 

Two  weights  are  connected  by  a  cord  passing  over  a  pulley  as  in  Atwood's 
machine,  friction  being  neglected,  and  also  the  masses  of  the  pulley  and  cord ; 
find:— 

84.  The  acceleration  when  one  weight  is  double  of  the  other. 

85.  The  acceleration  when  one  weight  is  to  the  other  as  20  to  21. 
Taking  g  as  980,  in  terms  of  the  cm.  and  sec.,  find  :— 

86.  The  velocity  acquired  in  10  sec.,  when  one  weight  is  to  the  other  as  39 
to  41. 

87.  The  velocity  acquired  in  moving  through  50  cm.,  when  the  weights  are  as 
19  to  21. 

88.  The  distance  through  which  the  same  weights  must  move  that  the  velocity 
acquired  may  be  double  that  in  last  question. 

89.  The  distance  through  which  two  weights  which  are  as  49  to  51  must  move 
that  they  may  acquire  a  velocity  of  98  cm.  per  sec. 


EXAMPLES.  245 

ENERGY  AND  WORK. 

90.  Express  in  ergs  the  kinetic  energy  of  a  mass  of  50  gm.  moving  with  a 
velocity  of  60  cm.  per  sec. 

91.  Express  in  ergs  the  work  done  in  raising  a  kilogram  through  a  height  of 
1  metre,  at  a  place  where  g  is  981. 

92.  A  mass  of  123  gm.  is  at  a  height  of  2000  cm.  above  a  level  floor.     Find  its 
energy  of  position  estimated  with  respect  to  the  floor  as  the  standard  level  (y 
being  981). 

93.  A  body  is  thrown  vertically  upwards  at  a  place  where  g  is  980.     If  the 
velocity  of  projection  is  9800  cm.  per  sec.  and  the  mass  of  the  body  is  22  gm., 
find  the  energy  of  the  body's  motion  when  it  has  ascended  half  way  to  its  maximum 
height.     Also  find  the  work  done  against  gravity  in  this  part  of  the  ascent. 

94.  The  height  of  an  inclined  plane  is  12  cm.,  and  the  length  24  cm.     Find 
the  work  done  by  gravity  upon  a  mass  of  1  gm.  in  sliding  down  this  plane  (g 
being  980),  and  the  velocity  with  which  the  body  will  reach  the  bottom  if  there 
be  no  friction. 

95.  If  the  plane  in  last  question  be  not  frictionless,  and  the  velocity  on 
reaching  the  bottom  be  20  cm.  per  sec.,  find  how  much  energy  is  consumed  in 
friction. 

96.  Find  the  work  expended  in  discharging  a  bullet  whose  mass  is  30  gm. 
with  a  velocity  of  40,000  cm.  per  sec.;  and  the  number  of  such  bullets  that  will 
be  discharged  with  this  velocity  in  a  minute  if  the  rate  of  working  is  7460 
million  ergs  per  sec.  (one  horse-power). 

97.  One  horse-power  being  defined  as  550  foot-pounds  per  sec. ;  show  that  it 
is  nearly  equivalent  to  8'8  cubic  ft.  of  water  lifted  1  ft.  high  per  sec.     (A  cubic 
foot  of  water  weighs  62£  Ibs.  nearly.     A  foot-pound  is  the  work  done  against 
gravity  in  lifting  a  pound  through  a  height  of  1  ft.) 

98.  How  many  cubic  feet  of  water  will  be  raised  in  one  hour  from  a  mine 
200  ft.  deep,  if  the  rate  of  pumping  be  15  horse-power? 

CENTRIFUGAL  FORCE. 

99.  "What  must  be  the  radius  of  curvature,  that  the  centrifugal  force  of  a 
body  travelling  at  30  miles  an  hour  may  be  one-tenth  of  the  weight  of  the  body ; 
g  being  981,  and  a  mile  an  hour  being  44'7  cm.  per  sec.? 

100.  A  heavy  particle  moves  freely  along  a  frictionless  tube  which  forms  a 
vertical  circle  of  radius  a.     Find  the  velocity  which  the  particle  will  have  at  the 
lowest  point,  if  it  all  but  comes  to  rest  at  the  highest.     Also  find  its  velocity  at 
the  lowest  point  if  in  passing  the  highest  point  it  exerts  no  pressure  against  the 
tube.     [Use  the  principle  that  what  is  lost  in  energy  of  position  is  gained  in 
energy  of  motion.] 

101.  Show  that  the  total  intensity  of  centrifugal  force  due    to  the  earth's 

rotation,  at  a  place  in   latitude  A,  is  w2  E  cos   X,  u  denoting  ^,   and  E  the 

earth's  radius ;  that  the  vertical  component  (tending  to  diminish  gravity)  is  w2 
E  cos2  A,  and  that  the  horizontal  component  (directed  from  the  pole  towards  the 
equator)  is  w2  E  cos  A  sin  A. 


24G  EXAMPLES. 

PENDULUM,  AND  MOMENT  OF  INERTIA. 

101*.  The  length  of  the  seconds  pendulum  at  Greenwich  is  99'413  cm.;  find 
the  length  of  a  pendulum  which  makes  a  single  vibration  in  lj  sec. 

102.  The  weight  of  a  fly-wheel  is  M  grammes,  and  the  distance  of  the  inside 
of  the  rim  from  the  axis  of  revolution  is  E  centims.     Supposing  this  distance  to 
be  identical  with  k  (§  117),  find  the  moment  of  inertia. 

If  a  force  of  F  dynes  acts  steadily  upon  the  wheel  at  an  arm  of  a  centims., 
what  will  be  the  value  of  the  angular  velocity  -™  after  the  lapse  of  t  seconds  from 
the  commencement  of  motion  ? 

103.  For  a  uniform  thin  rod  of  length  a,  swinging  about  a  point  of  suspension 
at  one  end,  the  moment  of  inertia  is  the  mass  of  the  rod  multiplied  by  |a2. 
Find  the  length  of  the  equivalent  simple  pendulum ;  also  the  moment  of  inertia 
round  a  parallel  axis  through  the  centre  of  the  rod. 

104.  At  what  point  in  its  length  must  the  rod  in  last  question  be  suspended 
to  give  a  minimum  time  of  vibration :  and  at  what  point  must  it  be  suspended  to 
give  the  same  time  of  vibration  as  if  suspended  at  one  end? 

105.  Show  that  if  P  be  the  mass  of  the  pulley  in  Atwood's  machine,  r  its 
radius,  and  P£2  its  moment  of  inertia,  the  value  of  C  in  §  100  will  be  P  JJ 
plus  the  mass  of  the  string.     [The  mass  of  the  friction-wheels  is  neglected.] 

106.  A  body  moves  with  constant  velocity  in  a  vertical  circle,  going  once 
round  per  second ;  and  its  shadow  is  cast  upon  level  ground  by  a  vertical  sun. 
Find  the  value  of  ft,  (§  111)  for  the  shadow,  using  the  centimetre  and  second  as 
units. 

107.  "What  is  the  value  of  ft  for  one  of  the  prongs  of  a  C  tuning-fork  which 
makes  512  complete  vibrations  per  second? 


PRESSURE  OF  LIQUIDS. 

Find,  in  gravitation  measure  (grammes  per  sq.  cm.),  atmospheric  pressure 
being  neglected : — 

108.  The  pressure  at  the  depth  of  a  kilometre  in  sea-water  of  density  1'025. 

109.  The  pressure  at  the  depth  of  65  cm.  in  mercury  of  density  13-59. 

110.  The  pressure  at  the  depth  of  2  cm.  in  mercury  of  density  13'59  sur- 
mounted by  3  cm.  of  water  of  unit  density,  and  this  again  by  li  cm.  of  oil  of 
density  '9. 

Find,  in  centimetres  of  mercury  of  density  13'6,  atmospheric  pressure  being 
included,  and  the  barometer  being  supposed  to  stand  at  70  cm.:— 

111.  The  pressure  at  the  depth  of  10  metres  in  water  of  unit  density. 

112.  The  pressure  at  the  depth  of  a  mile  in  sea-water  of  density  1-026  a  mile 
being  160933  cm. 

Find,  in  dynes  per  square  centimetre,  taking  g  as  981  :  — 

113.  The  pressure  due  to  1  cm.  of  mercury  of  density  13-596. 


EXAMPLES.  247 

114.  The  pressm-e  due   to  a  foot  of  water  of  unit  density,  a  foot  being 
30-48  cm. 

115.  The  pressure  due  to  the  weight  of  a  layer  a  metre  thick,  of  air  of  density 
•00129. 

116.  At  what  depth,  in  brine  of  density  I'l,  is  the  pressure  the  same  as  at  a 
depth  of  33  feet  in  water  of  unit  density  ? 

117.  At  what  depth,  in  oil  of  density  '9,  is  the  pressure  the  same  as  at  the 
depth  of  10  inches  in  mercury  of  density  13'596? 

118.  With  what  value  of  g  will  the  pressure  of  3  cm.  of  mercury  of  density 
13-596  be  4  x  1041 

Find,  in  grammes  weight,  the  amount  of  pressure  (atmospheric  pressure  being 
neglected) : — 

119.  On  a  triangular  area  of  9  sq.  cm.  immersed  in  naphtha  of  density  '848; 
the  centre  of  gravity  of  the  triangle  being  at  the  depth  of  6  cm. 

120.  On  a  rectangular  area  12  cm.  long,  and  9  cm.  broad,  immersed  in  mercury 
of  density  13-596 ;  its  highest  and  lowest  corners  being  at  depths  of  3  cm.  and  7 
cm.  respectively. 

121.  On  a  circular  area  of  10  cm.  radius,  immersed  in  alcohol  of  density  "791, 
the  centre  of  the  circle  being  at  the  depth  of  4  cm. 

122.  On  a  triangle  whose  base  is  5  cm.  and  altitude  6  cm.,  the  base  being  at 
the  uniform  depth  of  9  cm.,  and  the  vertex  at  the  depth  of  7  cm.,  in  water  of  unit 
density. 

123.  On  a  sphere  of  radius  r  centimetres,  completely  immersed  in  a  liquid  of 
density  d;  the  centre  of  the  sphere  being  at  the  depth  of  h  centimetres.     [The 
amount  of  pressure  in  this  case  is  not  the  resultant  pressure.] 


DENSITY,  AND  PRINCIPLE  OF  ARCHIMEDES. 
Densities  are  to  le  expressed  in  grammes  per  cullc  centimetre. 

124.  A  rectangular  block  of  stone  measures  86  x  37  x  16  cm.,  and  weighs 
120  kilogrammes.     Find  its  density. 

125.  A  specific-gravity  bottle  holds  100  gm.  of  water,  and  180  gm.  of  sulphuric 
acid.     Find  the  density  of  the  acid. 

126.  A  certain  volume  of  mercury  of  density  13'6  weighs  216  gm.,  and  the 
same  volume  of  another  liquid  weighs  14'8  gm.     Find  the  density  of  this  liquid. 

127.  Find  the  mean  section  of  a  tube  16  cm.  long,  which  holds  1  gm.  of  mercury 
of  density  13'6. 

128.  A  bottle  filled  with  water,  weighs  212  gm.    Fifty  grammes  of  filings  are 
thrown  in,  and  the  water  which  flows  over  is  removed,  still  leaving  the  bottle 
just  filled.     The  bottle  then  weighs  254  gm.     Find  the  density  of  the  filings. 

129.  Find  the  density  of  a  body  which  weighs  58  gm.  in  air,  and  46  gm.  in 
water  of  unit  density. 

130.  Find  the  density  of  a  body  which  weighs  63  gm.  in  air,  and  35  gm.  in  a 
liquid  of  density  "85. 


248  EXAMPLES. 

131.  A  glass  ball  loses  33  gm.  when  weighed  in  water,  and  loses  G  gm.  more 
when  weighed  in  a  saline  solution.     Find  the  density  of  the  solution. 

132.  A  body,  lighter  than  water,  weighs  102  gm.  in  air;  and  when  it  is  im- 
mersed in  water  by  the  aid  of  a  sinker,  the  joint  weight  is  23  gm.     The  sinker 
alone  weighs  50  gm.  in  water.     Find  the  density  of  the  body. 

133.  A  piece  of  iron,  when  plunged  in  a  vessel  full  of  water,  makes  10  grammes 
ran  over.     When  placed   in  a  vessel  full  of  mercury  it  floats,  displacing   78 
grammes   of  mercury.     Required  the  weight,   volume,  and  specific  gravity  of 
the  iron. 

134.  Find  the  volume  of  a  solid  which  weighs  357  gm.  in  air,  and  253  gm.  in 
water  of  unit  density. 

135.  Find  the  volume  of  a  solid  which  weighs  458  gm.  in  air,  and  409  gm.  in 
brine  of  density  1'2. 

136.  How  much  weight  will  a  body  whose  volume  is  47  cubic  cm.  lose,  by 
weighing  in  a  liquid  whose  density  is  2'5? 

137.  Find  the  weights  in  air,  in  water,  and  in  mercury,  of  a  cubic  cm.  of  gold 
of  density  19'3. 

138.  A  wire  1293  cm.  long  loses  508  gm.  by  weighing  in  water.     Find  its 
mean  section,  and  mean  radius. 

139.  A  copper  wire  2156  cm.  long  weighs  158  gm.  in  air,  and  140  gm.  in 
water.     Find  its  volume,  density,  mean  section,  and  mean  radius. 

140.  What  will  be  the  weights,  in  air  and  in  water,  of  an  iron  wire  1000  cm. 
long  and  a  millimetre  in  diameter,  its  density  being  7'7? 

141.  How  "much  water  will  be  displaced  by  1000  c.c.  of  oak  of  density  '9, 
floating  in  equilibrium] 

142.  A  ball,  of  density  20  and  volume  3  c.c.,  is  surmounted  by  a  cylindrical 
stem,  of  density  2'5,  of  length  12  cm.,  and  of  cross  section  |  sq.  cm.     What  length 
of  the  stem  will  be  in  air  when  the  body  floats  in  equilibrium  in  mercury  of 
density  13'6l 

143.  A  hollow  closed  cylinder,  of  mean  density  '4  (including  the  hollow  space), 
is  weighted  with  a  ball  of  volume  5,  and  mean  density  2.     What  must  be  the 
volume  of  the  cylinder,  that  exactly  half  of  it  may  be  immersed,  when  the  body 
is  left  to  itself  in  water? 

144.  A  long  cylindrical  tube,  constructed  of  flint  glass  of  density  3,  is  closed 
at  both  ends,  and  is  found  to  have  the  property  of  remaining  at  whatever  depth 
it  is  placed  in  water.     If  the  mass  of  the  ends  can  be  neglected,  show  that  the 
ratio  of  the  internal  to  the  external  radius  is  */  -r- 

145.  A  glass  bottle  provided  with  a  stopper  of  the  same  material  weighs  120 
gm.  when  empty.     When  it  is  immersed  in  water,  its  apparent  weight  is  10  gm., 
but  when  the  stopper  is  loosened  and  the  water  let  in,  its  apparent  veight  is  80 
gm.     Find  the  density  of  the  glass  and  the'capacity  of  the  bottle. 

146.  A  hydrometer  sinks  to  a  certain  depth  in  a  fluid  of  density  -8;  and  if 
100  gm.  be  placed  upon  it,  it  sinks  to  the  same  depth  in  water.     Find  the  weight 
of  the  hydrometer. 

147.  Find  the  mean  density  of  a  combination  of  8  parts  by  volume  of  a  sub- 
stance of  density  7,  with  19  of  a  substance  of  density  3. 


EXAMPLES.  249 

148.  Find  the  mean  density  of  a  combination  of  8  parts  by  weight  of  a  sub- 
stance of  density  7,  with  19  of  a  substance  of  density  3. 

149.  "What  volume  of  fir,  of  density  '5,  must  be  joined  to  3  c.c.  of  iron,  of 
density  7'1,  that  the  mean  density  of  the  whole  may  be  unity? 

150.  What  mass  of  fir,  of  density  '5,  must  be  joined  to  300  gm.  of  iron,  of 
density  7'1,  that  the  mean  density  of  the  whole  may  be  unity? 

151.  Two  parts  by  volume  of  a  liquid  of  density  '8,  are  mixed  with  7  of  water, 
and  the  mixture  shrinks  in  the  ratio  of  21  to  20.     Find  its  density. 

152.  A  piece  of  iron  of  density  7'5  floats  in  mercury  of  density  13'5,  and  is 
completely  covered  with  water  which  rests  on  the  top  of  the  mercury.     How  much 
of  the  iron  is  immersed  in  the  mercury? 

153.  Two  liquids  are  mixed.     The  total  volume  is  3  litres,  with  a  sp.  gr.  of 
0-9.     The  sp.  gr.  of  the  first  liquid  is  1'3,  of  the  second  07.     Find  their  volumes. 

154.  What  volume  of  platinum  of  density  21 '5  must  be  attached  to  a  litre  of 
iron  of  density  7'5  that  the  system  may  float  freely  at  all  depths  in  mercury  of 
density  13'5? 

155.  What  must  be  the  thickness  of  a  hollow  sphere  of  platinum  with  an  ex- 
ternal radius  of  1  decim.,  that  it  may  barely  float  in  water? 

156.  A  sphere  of  cork  of  density  "24,  3  cm.  in  radius,  is  weighted  with  a 
sphere  of  gold  of  density  19'3.    What  must  be  the  radius  of  the  latter  that  the 
system  may  barely  float  in  alcohol  of  density  '8? 

157.  An  alloy  of  gold  and  silver  has  density  D.     The  density  of  gold  is  d,  that 
of  silver  d'.     Find  the  proportions  by  weight  of  the  two  metals  in  the  alloy,  sup- 
posing that  neither  expansion  nor  contraction  occurs  in  its  formation. 

158.  A  mixture  of  gold,  of  density  19'3,  with  silver,  of  density  10'5,  has  the 
density  18.     Assuming  that  the  volume  of  the  alloy  is  the  sum  of  the  volumes  of 
its  components,  find  how  many  parts  of  gold  it  contains  for  one  of  silver— (a)  by 
volume;  (b)  by  weight. 

159.  A  body  weighs  t^M  dynes  in  air  of  density  A,  gm  in  water,  and  gx  in 
vacuo.     Find  x  in  terms  of  M,  m,  and  A. 


CAPILLARITY. 

1GO.  A  horizontal  disc  of  glass  is  held  up  by  means  of  a  film  of  water  between 
it  and  a  similar  disc  of  the  same  or  a  larger  size  above  it. 
If  E,  denote  the  radius  of  the  lower  disc, 

d  the  distance  between  the  discs,  which  is  very  small  compared  with  E, 
T  the  surface  tension  of  water, 

show  that  the  weight  of  the  lower  disc  together  with  that  of  the  water 

between  the  discs  is  approximately  equal  to  — -j —  • 

[The  disc  of  water  will  be  concave  at  the  edge,  and  the  radius  of  curvature  of 
the  concavity  may  be  taken  as  %d.] 

161.  The  surface-tension  of  water  at  20°  C.  is  81  dynes  per  linear  centim. 
How  high  will  water  be  elevated  by  capillary  action  in  a  wetted  tube  whose  dia- 
meter is  half  a  millimetre? 


250  EXAMPLES. 

162.  Uow  much  will  mercury  be  depressed  by  capillary  action  in  a  glass  tube 
of  half  a  millimetre  diameter,  the  surface-tension  of  mercury  at  20  C.  being  41 
dynes  per  cm.,  its  density  13'54,  and  the  cosine  of  the  angle  of  contact  '703 

163    Show  by  the  method  of  §  186  that  the  capillary  elevation  or  depression 
will  be  the  same  in  a  square  tube  as  in  a  circular  tube  whose  diameter  is  equal 
a  side  of  the  square. 

164.  Two  equal  discs  in  a  vertical  position  have  a  film  of  water  between 
them  sustained  by  capillary  action.     Show  that  if  the  water  at  the  lowest  point 
is  at  atmospheric  pressure,  the  water  at  the  centre  of  the  discs  is  at  a  pressure  less 
than  atmospheric  by  rg  dynes  per  sq.  cm.,  r  being  the  common  radius  of  the 
discs  in  cm.;  and  that  the  discs  are  pressed  together  with  a  force  of  *  1*g  dynes. 

BAROMETER,  ASD  BOYLE'S  LAW. 

165.  A  bent  tube,  having  one  end  open  and  the  other  closed,  contains  mercury 
which  stands  20  cm.  higher  in  the  open  than  in  the  closed  branch.     Compare  the 
pressure  of  the  air  in  the  closed  branch  with  that  of  the  external  air;  the  baro- 
meter at  the  time  standing  at  75  cm. 

166.  The  cross  sections  of  the  open  and  closed  branches  of  a  siphon  barometer 
are  as  6  to  1.     What  distance  will  the  mercury  move  in  the  closed  branch,  when 
a  normal  barometer  alters  its  reading  by  1  inch? 

167.  If  the  section  of  the  closed  limb  of  a  siphon  barometer  is  to  that  of  the 
open  limb  as  a  to  b,  show  that  a  rise  of  1  cm.  in  the  mercury  in  the  closed  limb 
corresponds  to  a  rise  of  ^^  cm.  of  the  theoretical  barometer. 

168.  Compute,  in  dynes  per  sq.  cm.,  the  pressure  due  to  the  weight  of  a  column 
of  mercury  76  cm.  high  at  the  equator,  where  g  is  978,  and  at  the  pole,  where  g 
is  983. 

169.  The  volimes  of  a  given  quantity  of  mercury  at  0°  C.  and  100°  C.  are  as 
1  to  1-0182.     Compute  the  height  of  a  column  of  mercury  at  100°,  which  will 
produce  the  same  pressure  as  76  cm.  of  mercury  at  0°. 

170.  The  volumes  of  a  given  mass  of  mercury,  at  0°  and  20°,  are  as  1  to  1'0036. 
Find  the  height  reduced  to  0°,  when  the  actual  height  (in  true  centimetres),  at  a 
temperature  of  20°,  is  76'2. 

171.  In  performing  the  Torricellian  experiment  a  little  air  is  left  above  the 
mercury.    If  this  air  expands  a  thousandfold,  what  difference  will  it  make  in  the 
height  of  the  column  of  mercury  sustained  when  a  normal  barometer  reads 
76  cm.? 

172.  In  performing  the  Torricellian  experiment,  an  inch  in  length  of  the  tube 
is  occupied  with  air  at  atmospheric  pressure,  before  the  tube  is  inverted.     After 
the  inversion,  this  air  expands  till  it  occupies  15  inches,  while  a  column  of 
mercury  28  inches  high  is  sustained  below  it.     Find  the  true  barometric  height. 

173.  The  mercury  stands  at  the  same  level  in  the  open  and  in  the  closed  branch 
of  a  bent  tube  of  uniform  section,  when  the  air  confined  at  the  closed  end  is  at 
the  pressure  of  30  inches  of  mercury,  which  is  the  same  as  the  pressure  of  the 
external  air.     Express,  in  atmospheres,  the  pressure  which,  acting  on  the  surface 
of  the  mercury  in  the  open  branch,  compresses  the  confined  air  to  half  its  original 


EXAMPLES.  251 

volume,  and  at  the  same  time  maintains  a  difference  of  5  inches  in  the  levels  of 
the  two  mercurial  columns. 

174.  At  what  pressure  (expressed  in  atmospheres)  will  common  air  have  the 
same  density  which  hydrogen  has  at  one  atmosphere ;  their  densities  when  com- 
pared at  the  same  pressure  being  as  1276  to  88'4? 

175.  Two  volumes  of  oxygen,  of  density  -00141,  are  mixed  with  three  of 
nitrogen,  of  density  '00124.     Find  the  density  of  the  mixture— (a)  if  it  occupies 
five  volumes;  (b)  if  it  is  reduced  to  four  volumes. 

176.  The  mass  of  a  cub.  cm.  of  air,  at  the  temperature  0°  C.,  and  at  the  pressure 
of  a  million  dynes  to  the  square  cm.,  is  '0012759  gramme.     Find  the  mass  of  a 
cubic  cm.  of  air  at  0°  C.,  under  the  pressure  of  76  cm.  of  mercury — (a)  at  the  pole, 
where  g  is  983-l;  (b)  at  the  equator,  where  g  is  978'1 ;  (c)  at  a  place  where  g  is 
981. 

177.  Show  that  the  density  of  air  at  a  given  temperature,  and  under  the 
pressure  of  a  given  column  of  mercury,  is  greater  at  the  pole  than  at  the  equator 
by  about  1  part  in  196 ;  and  that  the  gravitating  force  of  a  given  volume  of  it 
is  greater  at  the  pole  than  at  the  equator  by  about  1  part  in  98. 

178.  A  cylindrical  test-tube,  1  decim.  long,  is  plunged,  mouth  downwards, 
into  mercury.     How  deep  must  it  be  plunged  that  the  volume  of  the  inclosed  air 
may  be  diminished  by  one-half? 

179.  The  pressure  indicated  by  a  siphon  barometer  whose  vacuum  is  defective 
is  750  mm.,  and  when  mercury  is  poured  into  the  open  branch  till  the  barometric 
chamber  is  reduced  to  half  its  former  volume,  the  pressure  indicated  is  740  mm. 
Deduce  the  true  pressure. 

180.  An  open  manometer,  formed  of  a  bent  tube  of  iron  whose  two  branches 
are  parallel  and  vertical,  and  of  a  glass  tube  of  larger  size,  contains  mercury  at 
the  same  level  in  both  branches,  this  level  being  higher  than  the  junction  of  the 
iron  with  the  glass  tube.     What  must  be  the  ratio  of  the  sections  of  the  two 
tubes,  that  the  mercury  may  ascend  half  a  metre  in  the  glass  tube  when  a  pres- 
sure of  6  atmospheres  is  exerted  in  the  opposite  branch? 

181.  A  curved  tube  has  two  vertical  legs,  one  having  a  section  of  1  sq.  cm.,  the 
other  of  10  sq.  cm.     "Water  is  poured  in,  and  stands  at  the  same  height  in  both 
legs.    A  piston,  weighing  5  kilogrammes,  is  then  allowed  to  descend,  and  press 
with  its  own  weight  upon  the  surface  of  the  liquid  in  the  larger  leg.     Find  the 
elevation  thus  produced  in  the  surface  of  the  liquid  in  the  smaller  leg. 

PUMPS,  &c. 

182.  The  sectional  area  of  the  small  plunger  in  a  Bramah  press  is  1  sq.  cm., 
and  that  of  the  larger  100  sq.  cm.     The  lever  handle  gives  a  mechanical  advan- 
tage of  6.     What  weight  will  the  large  plunger  sustain  when  1  cwt.  is  hung  from 
the  handle? 

183.  The  diameter  of  the  small  plunger  is  half  an  inch;  that  of  the  larger 
1  foot.     The  arms  of  the  lever  handle  are  3  in.  and  2  ft.     Find  the  total  mechan- 
ical advantage. 

184.  Find,  in  grammes  weight,  the  force  required  to  sustain  the  piston  of  a 
suction-pump  without  friction,  if  the  radius  of  the  piston  be  15  cm.,  the  depth 


252  EXAMPLES. 

from  it  to  the  surface  of  the  water  in  the  well  600  cm.,  and  the  height  of  the 
column  of  water  above  it  50  cm.  Show  that  the  answer  does  not  depend  on  the 
size  of  the  pipe  which  leads  down  to  the  well. 

185.  Two  vessels  of  water  are  connected  by  a  siphon.     A  certain  point  P  in 
its  interior  is  10  cm.  and  30  cm.  respectively  above  the  levels  of  the  liquid  in  the 
two  vessels.     The  pressure  of  the  atmosphere  is  1000  grammes  weight  per  sq.  cm. 
Find  the  pressure  which  will  exist  at  P— (a)  if  the  end  which  dips  in  the  upper 
vessel  be  plugged ;  (6)  if  the  end  which  dips  in  the  lower  vessel  be  plugged. 

186.  If  the  receiver  has  double  the  volume  of  the  barrel,  find  the  density  of 
the  air  remaining  after  10  strokes,  neglecting  leakage,  &c. 

187.  Air  is  forced  into  a  vessel  by  a  compression  pump  whose  barrel  has  j^th 
of  the  volume  of  the  vessel.     Compute  the  density  of  the  air  in  the  vessel  after 
20  strokes. 

188.  In  the  pump  of  Fig.  136  show  that  the  excess  of  the  pressure  on  the  upper 
above  that  on  the  lower  side  of  the  piston,  at  the  end  of  the  first  up-stroke,  is 

•y 

y — y-,  of  an  atmosphere  [in  the  notation  of  §  230];  and  hence  that  the  first 
stroke  is  more  laborious  with  a  small  than  with  a  large  receiver. 

189.  In  Tate's  pump  show  that  the  pressure  to  be  overcome  in  the  first  stroke 
is  nearly  equal  to  an  atmosphere  during  the  greater  part  of  the  stroke ;  and  that, 
when  half  the  air  has  been  expelled  from  the  receiver,  the  pressure  to  be  over- 
come varies,  in  different  parts  of  the  stroke,  from  half  an  atmosphere  to  an  atmo- 
sphere. 


ANSWEES  TO  EXAMPLES. 

Ex.  1.  14-14.  Ex.  2.  13.  Ex.  3.  7'07  each.  Ex.  4.  10.  Ex.  5.  141-4. 
Ex.  6.  70'7  each.  Ex.  7.  Introduce  a  force  equal  and  opposite  to  the  resultant. 
Then  we  have  three  forces  making  angles  of  120°  with  each  other.  Ex.  9.  Equal 
to  one  of  the  forces. 

Ex.  10*.  28.  Ex.  11*.  40  in.  from  smaller  weight.  Ex.  12.  60  Ibs.  by  A, 
40  Ibs.  by  B.  Ex.  13.  2?  Ibs.  Ex.  14.  2  Ibs.  Ex.  15.  15  in.  from  centre. 
Ex.  16.  121  Ibs.  Ex.  17.  32  Ibs.  Ex.  18.  10'4  ft.  nearly.  Ex.  19.  6  Ibs.  Ex. 
20.  2?  ft.  from  end.  Ex.  21.  2}f.  Ex.  22.  2  units  acting  at  distance  of  5  yards 
trom  the  greater  force.  Ex.  23.  6  ft.  from  the  end  ;  pressure  2  units.  Ex.  24. 
4|  Ibs  Ex.  25.  2fr  Ibs,  lOf  Ibs.  Ex.  26.  f  in.  Ex.  27.  2ft-  in.  Ex.  28.  I  of 
side  of  square.  Ex.  29.  &  of  diagonal  of  large  square.  Ex.  30.  wVV  cm.  from 

+T+pshere'  Ex'  3L  Denoting  side  of  s(iuare  by  a>  distance  from  AZ 

distance  froin 


Ex.  34.  4&  cm.     Ex.  35.  5&  cm.      Ex.  3G.  17.     Ex.  37.    -1,  0,  +  18.     Ex. 
38.  i\V  (V(&2  +  c*)-c).     Ex.  39.  fWL     Ex.  40.  a  (II-/,)*. 


EXAMPLES.  253 

Ex.  44.  14  Ibs.  Ex.  45.  (a)  133  J  Ibs.;  (6)  166f  Ibs.  Ex.  46.  1  to  603  nearly. 
Ex.  47.  373J. 

Ex.  49.  50.  Ex.  50.  72.  Ex.  61.  7  cm.  per  sec.  Ex.  52.  35.  Ex.  53.  60. 
Ex.  54.  60  dynes.  Ex.  55.  6  dynes.  Ex.  56.  7  dynes.  Ex.  57.  Smaller  mass 
rfa,  larger  y^s  cm.  per  sec.  Ex.  58.  Inversely  as  masses  of  bullet  and  gun. 
Ex.  59.  Mass  of  bullet  is  j^  of  mass  of  block. 

Ex.  60.  98  cm.  per  sec.  Ex.  61.  4'9  cm.  Ex.  62.  490  cm.  Ex.  63.  7  sec. 
Ex.  64.  122^  cm.  Ex.  65.  626  cm.  per  sec.  Ex.  66.  6  cm.  per  sec.  upwards. 
Ex.  67.  45'9  cm.  above  point  of  projection.  Ex.  68.  1650  cm.  per  sec.  downwards. 
Ex.  69.  1062i  cm.  below  starting  point.  Ex.  70.  384  ft,  per  sec.  Ex.  71.  2304  ft. 
Ex.  72.  1&  ft.  Ex.  73.  5  sec.  Ex.  74.  16  ft.  Ex.  75.  40  ft.  per  sec.  Ex.  76. 
4  ft.  per  sec.  upwards.  Ex.  77.  156  ft.  Ex.  78.  78  ft.  per  sec.  Ex.  79.  81  ft. 
Ex.  80.  After  4'52  sec.  At  135'6  m.  from  tower.  Ex.  81.  After  -41  sec.  from 
dropping  of  second  body. 

Ex.  84.  £  g.  Ex.  85.  ^  g.  Ex.  86.  245  cm.  per  sec.  Ex.  87.  70  cm.  per 
sec.  Ex.  88.  200  cm.  Ex.  89.  245  cm. 

Ex.  90.  90,000  ergs.  Ex.  91.  98,100,000  ergs.  Ex.  92.  241,326,000  ergs. 
Ex.  93.  528,220,000  ergs  each.  Ex.  94.  11,760  ergs ;  V  23520  =  153'4  cm.  per  sec. 
Ex.  95.  11,560  ergs.  Ex.  96.  24  x  109  ergs  in  each  discharge.  Not  quite  19 
discharges  per  min.  Ex.  98.  2376  nearly. 

Ex.  99.  18330  cm.  or  about  600  ft.     Ex.  100.  2  V"^a",  V5^*7 
Ex.  101*.  223-679  cm.     Ex.  102.  ME2,  ™~     Ex.  103.  fa;  mass  of  rod  multi- 
plied by  ^a2.     Ex.  104.  At  either  of  the  two  points  distant  -p  from  centre ;  at 

either  of  the  two  points  distant  -j  from  centre.    Ex.  106.  (£^)2  =  39-48.    Ex.  107. 
(1024  w)2=  10350000. 


Ex.  108.  102500.  Ex.  109.  883'35.  Ex.  110.  31-53.  Ex.  111.  149'5.  Ex. 
112.  12217.  Ex.  113.  13338.  Ex.  114.  29901.  Ex.  115.  126'5.  Ex.  116.  30. 
Ex.  117.  12  ft.  7  in.  Ex.  118.  980-68.  Ex.  119.  4579.  Ex.  120.  7342.  Ex. 
121.  994.  Ex.  122.  125.  Ex.  123.  4*r2M 

Ex.  124.  2-357.  Ex.  125.  1'8.  Ex.  126.  '932.  Ex.  127.  '0046  sq.  cm.  Ex. 
128.  6-25.  Ex.  129.  4|.  Ex.  130.  1'9125.  Ex.  131.  1ft.  Ex.  132.  f|.  Ex. 
133.  10  cub.  cm.,  78  gra.,  7'8.  Ex.  134.  104.  Ex.  135.  40'83.  Ex.  136.  117'5. 
Ex.  137.  19-3,  18-3,  57.  Ex.  138.  '393  sq.  cm.,  '354  cm.  Ex.  139.  18,  8777", 
•00835  sq.  cm.,  '0516  cm.  Ex.  140.  60'48,  52'62.  Ex.  141.  900  c.c.  Ex.  142. 
5-56  cm.  Ex.  143.  50  c.c.  Ex.  145.  3,  70  c.c.  Ex.  146.  400  gm.  Ex.  147. 
4^V  =  4-185.  Ex.  148.  3ft6T  =  3'6115.  Ex.  149.  36'6  c.c.  Ex.  150.  2577  gm. 
Ex.  151.  1-0033.  Ex.  152.  if  of  the  iron.  Ex.  153.  1  lit.  of  first,  2  lit.  of  second. 

Ex.  154.  |  of  a  litre.    Ex.  155.  !-%/*-  decim.  =  -158  cm.     Ex.  156.  ^4^1  = 

v  4d  18  D 

•935  cm.     Ex.  157.  Gold:  silver  :  :  --^ ;:  g-J-     Ex.  158.  (a)  577,  (6)  10'6. 
Ex.  159.  M,""!A- 


254  EXAMPLES. 

Ex.  161.  6'6  cm.  nearly.     Ex.  162.  1*77  cm. 

Ex.  165.  if.  Ex.  166.  f  in.  Ex.  168.  1010564,  1015730.  Ex.  169.  77'3832. 
Ex.  170.  75-93.  Ex.  171.  '076.  Ex.  172.  30  in.  Ex.  173.  2J.  Ex.  174.  -0693. 
Ex.  175.  (a)  -001308,  (b)  '001635.  Ex.  176.  (a)  '0012961,  (b)  '0012895,  (c)  '0012933. 
Ex.  177.  d  varies  as  g,  and  therefore  gd  varies  as  #2.  Ex.  178.  Its  top  must  be 
76-5  =  71  cm.  deep.  Ex.  179.  760  m.  Ex.  180.  33  to  5.  Ex.  181.  454j6r  cm. 

Ex.  182.  30  tons.  Ex.  183.  4608.  Ex.  184.  459500  nearly.  Ex.  185.  (a)  970. 
(b)  990  gin.  wt.  per  sq.  cm.  Ex.  186.  ^  of  an  atmosphere,  nearly.  Ex.  187.  3 
atmospheres. 


INDEX  TO  PAKT  I. 


Absorption  of  gases,  177. 

Charts  of  weather,  163. 

Errors  and  corrections,  signs  of, 

Acceleration  defined,  51. 

Circular  motion,  59. 

151- 

Air,  weight  of,  138,  139. 

Clearance,  see  untraversed  space, 

Exhaustion,  limit  of,  188. 

—  pump,  179. 

189,  213. 

—  ,  rate  of,  180. 

—  chamber,  218. 

Coefficients  of  elasticity,  79. 

Expansibility  of  gases,  137. 

—  film,  adherent,  177. 

—  of  friction,  8  1. 

Alcoholimeters,  114,  115. 

Colloids,  135. 

Fahrenheit's  barometer,  156. 

Am-plitude  of  vibration,  63. 

Communicating  vessels,  118,  125. 

—  hydrometer,  in. 

Aneroid,  154. 

Component  along  a  line,  16. 

Fall  in  vacuo,  49. 

Annual  and  diurnal  variations,  161. 

Components,  7. 

Falling  bodies,  52. 

Archimedes'  principle,  97. 

Compressed-air  machines,  202. 

Film  of  air  on  solids,  177. 

Aristotle's  experiment,  138. 

Compressibility,  79. 

Films,  tension  in,  127-130,  133,  134. 

Arithmetical  lever,  12. 

Compressing  pump,  199. 

Fire-engine,  218. 

Ascent  in    capillary  tubes,    124, 

Conservation  of  energy,  74-76. 

Float-adjustment    of    barometer, 

125,  128. 

Constant  load,  weighing  with,  37. 

147. 

Atmosphere,  140. 

Contracted  vein,  225. 

Floatation,  102. 

—  standard  of  pressure,  141. 

Contractile  film,  127-130,  133,  134. 

Floating  needles,  103. 

Attractions,  apparent,  133. 

Convertibility  of  centres,  70. 

Fluid,  perfect,  83. 

Atwood's  machine,  57. 

Corrections  of  barometer,  148-151. 

Force,  3. 

Axis  of  couple,  14. 

Counterpoised  barometer,  155. 

—  ,  amount  of,  44. 

—  of  wrench,  15. 

Couple,  13. 

—  ,  intensity  of,  44. 

Crystalloids,  135. 

—  ,  unit  of,  44,  48. 

Babinet's  air-pump,  196. 
Back-pressure  on  discharging  ves- 

Cupped-leather collar,  222. 
Cycloidal  pendulum,  67. 

Forcing-pump,  216. 
Fortin's  barometer,  144. 

sel,  92,  225,  226. 

Cyclones,  165. 

Fountain  in  vacuo,  187. 

Balance,  34-40. 

—  ,  intermittent,  230. 

Balloons,  204-208. 

D'Alembert's  principle,  96. 

Free-piston  air-pump,  196. 

Barker's  mill,  93. 

Deflecting  force,  60. 

Friction,  81,  82. 

Barographs,  156,  158. 

Deleuil's  air-pump,  196. 

—  in  connection  with  conservation 

Barometer,  142. 

Density,  absolute  and  relative,  105. 

of  energy,  76. 

—  ,  corrections  of,  148-151. 

—  ,  determination  of,  106—112. 

Froude  on  contracted  vein,  225. 

Barometric       measurement       of 

—  ,  table  of,  xii. 

heights,  159-161. 

Depression,   capillary,    124,    125, 

Galileo  on  falling  bodies,  49. 

—  prediction,  163. 

128. 

—  on  suction  by  pumps,  142. 

Baroscope,  204. 

Despretz's  experiments  on  Boyle'? 

Gases,  expansibility  of,  137. 

Beaume's  hydrometers,  113. 

law,  168. 

Geissler's  air-pump,  191. 

Bianchi's  air-pump,  183. 

Dialysis,  135. 

Geometric    decrease  of  pressure 

Bladder,  burst,  187. 

Diameters,  law  of,  125. 

upwards,  160. 

Bourdon's  gauge,  175. 

Diffusion,  135. 

Gimbals,  147. 

Boyle's  law,  166. 

Displaced  liquid  defined,  100. 

Gradient,  barometric,  164. 

—  tube,  166. 

Diurnal  barometric  curve,  161. 

Gramme,  105. 

Bramah  press,  222. 

Diver,  Cartesian,  101. 

Graphical  interpolation,  116. 

Bubbles  filled  with  hydrogen,  205. 
—  ,  tension  and  pressure  in,  130. 

Double-acting  pumps,  183,  218. 
Double-barrelled  air-pump,  i8t. 

Gravesande's  apparatus,  7. 
Gravitation  units  of  force,  4,  106. 

Buoyancy,  centre  of,  98,  100. 

Double  exhaustion,  194. 

Gravity,  apparent  and  true,  61. 

Buys  Ballot's  law,  164. 

—  weighing,  35. 

—  ,  centre  of,  17-21. 

Drops,  131. 

,  its  velocity,  46. 

Caissons,  202. 

—  ,  formula  for  its  intensity,  51. 

Camphor,  movements  of,  134. 

Dynamometer,  4. 

—  measured  by  pendulums,  72. 

Capillarity,  124-134. 

Dyne,  48. 

—  proportional  to  mass,  50. 

Cartesian  diver,  101. 

Guinea-and-feather     experiment. 

Cathetometer,  144. 

Efficiency  of  pumps,  214. 

49- 

Centre  of  buoyancy,  98,  100. 

Efflux  of  liquids,  224. 

—  of  gravity,  I7-2i. 
by  experiment,  22. 

from  air-tight  spaces,  229. 
Egg  in  water,  100. 

Head  of  liquid,  224. 
Heights  measured  by  barometer, 

,  velocity  of,  46. 

Elasticity,  77-80. 

159-161. 

—  of  mass,  47. 

Elevation,  capillary,  124-128. 

Hemispheres,  Magdeburg,  187. 

—  of  oscillation,  71. 

Endosmose,  134. 

Homogeneous   atmosphere,    159, 

—  of  parallel  forces,  10,  17. 

Energy,  conservation  of,  74-76. 

1  60. 

—  of  pressure,  93. 

Energy,  kinetic,  73. 

"  Horizontal  "  defined,  17. 

Centrifugal  force,  60,  95. 

—  ,  static  or  potential,  73. 

Horse-power,  xi. 

—  pump,  219. 

English  air-pumps,  184. 

Hydraulic  press,  87,  221. 

C.G.S.  system,  48. 

Equilibrium,  4. 

—  tourniquet,  93. 

Change  of  momentum,  42. 
—  of  motion,  42. 

Equivalent  simple  pendulum,  66. 
Erg,  48. 

Hydrodynamics,  83. 
Hydrogen,  bubbles  filled  with,  205. 

256 


INDEX   TO   PART   I. 


Hydrokinetics,  83. 
Hydrometers,  110-117. 

CErsted's  piezometer,  79. 
Oscillation,  centre  of,  71. 

Siphon,  231. 
—  for  sulphuric  acid,  234. 
Siphon-barometer,  151. 

Hydrostatics,  83. 
Hypsometric  formula,  161. 

Parachute,  207. 
Paradox,  hydrostatic,  91. 

Specific  gravity,  105. 
by  weighing  in  water,  108. 

Immersed  bodies,  98. 
Inclined  plane,  32. 
Index  errors  and  corrections,  151. 

Parallel  forces,  9-14. 
Parallelogram  of  forces,  7,  43. 
—  of  velocities,  43. 
Parallelepiped  of  forces,  8. 

flask,  107. 
,  table  of,  xii. 
Spirit-levels,  120-123. 
Sprengel's  air-pump,  193. 

u  ,  4  . 

Pascal's  mountain  experiment,  142. 

Spring-balance,  4. 

—  ,  moment  ot,  DO. 
Inexhaustible  bottle,  230. 
Insects  walking  on  water,  104. 
Intermittent  fountain,  230. 

—  principle,  86. 
—  vases,  89. 
Pendulum,  62. 
—  ,  compound,  70. 

Stability,  21-28,  38. 
Standard  kilogramme,  105. 
Statics,  4- 
Steelyard,  40. 

Isochronous  vibrations,  66,  78. 

—  ,  cycloidal,  67. 
—  ,  isochronism  of,  64. 

Suction,  211. 
—  pump,  211. 

Jet-pump,  219. 
Jets,  liquid,  224. 

—  ,  simple,  62. 
—  ,  time  of  vibration  of,  65. 
Period  of  vibration,  63. 

Sugar-boiling,  202. 
Superposed  liquids,  83. 
Surface  of  liquids  level,  85. 

Kater's  pendulum,  71. 

"  Perpetual  motion,"  26. 

Surface-tension,  127-130,  133,  134. 

Kinetic  energy,  73. 

Phial  of  four  elements,  89. 

,  table  of,  134. 

Kinetics,  4. 

Photographic  registration,  157. 

King's  barograph,  155,  156. 

Piezometer,  79. 

Tantalus'  cup,  285. 

Kravogl's  air-pump,  190. 

Pile-driving,  75 

Torricellian  experiment,  141. 

Laws  of  motion,  41-45. 
Levels,  119-123. 

Plateau's  experiments,  131. 
Platinum     causing     igniiion     of 

Torricelli's  theorem  on  efflux,  224. 
Tourniquet,  hydraulic,  93. 

Lever,  29. 
Limit  to  action  of  air-pump,  188. 
Liquids  find  their  own  level,  118. 

hydrogen,  177. 
Plunger,  216. 
Pneumatic  despatch,  202. 

Trajectory,  54. 
Translation  and  rotation,  3. 
Transmission  of  pressure  in  fluids, 

—  in  superposition,  88. 
Magdeburg  hemispheres,  187. 

Potential  energy,  73. 
Pressure,  centre  of,  93. 
—  ,  hydrostatic,  84. 

86. 
Triangle  of  forces,  6. 
Twaddell's  hydrometer,  114. 

Magic  funnel,  229. 
Manometers,  172-175. 
Marine  barometer,  153. 

—  ,  intensity  of,  83. 
—  on  immersed  surfaces,  93. 
—  ,  reduction  of,  to  absolute  mea- 

Uniform  acceleration,  50. 
Unit  of  force,  44,  48. 

Mariotte's  bottle,  235. 

—  law,  166. 
—  tube,  166. 
Mass,  44,  45. 
—  and  gravitation   proportional, 
50- 

Pressure-gauges,  172-175. 
Pressure-height  defined,  159,  220. 
Pressure  in  air  computed,  160. 
—  least  where  velocity  is  greatest, 

Units  of  measurement,  47. 
—  ,  C.G.S.,  48. 
Unstable  equilibrium,  21-28,  38. 
Untraversed  space,  189,  213. 
Upward  pressure  in  liquids,  88. 

—  ,  centre  of,  47. 
Mechanical  advantage,  30. 

Principle  of  Archimedes,  97. 
Projectiles,  53. 

Vena  contracta,  225. 
Vernier,  145. 

—  powers,  29-33. 
Mechanics,  2. 
Meniscus,  131. 

Pulleys,  31. 
Pump,  forcing,  216. 
—  ,  suction,  211. 

"Vertical  "denned,  17. 
Vessels    in    communication,    118, 
125. 

Metacentre,  103. 

Pumps,  efficiency  of,  214. 

Vibrations,  66. 

Metallic  barometer,  155. 

—  ,  when  small,  isochronous,  78. 

Mixtures,  density  of,  115. 
—  of  gases,  176. 

Quantity  of  matter,  45. 

Volumes  measured  by  weighing  in 

Moduli  of  elasticity,  78. 

Range  and  amplitude,  161. 

' 

Moment  of  couple,  13. 

Rarefaction,  limit  of,  188. 

Water,  compressibility  of,  79. 

—  of  force  about  point,  n. 

—  ,  rate  of,  180. 

—  level,  119. 

—  of  inertia,  68. 

Reaction,  4,  15,  45. 

—  supply  of  towns,  118. 

Momentum,  44. 

—  of  issuing  jet,  92,  225,  226. 

Wedge,  33- 

Morin's  apparatus,  55. 

Rectangular  components,  15. 

Weighing,  double,  35. 

Motion,  laws  of,  41-45. 

Regnault's  experiments  on  Boyle's 

—  in  water,  108. 

Motions,  composition  of,  42. 

law,  169-172. 

—  with  constant  load,  37. 

Mountain-barometer,    theory    of, 

Resistance  of  the  air,  49,  53. 

Weight  affected  by  air,  209. 

159-161. 

Resolution,  15. 

"Weight"  ambiguous,  106. 

Multiple-tube  barometer,  157. 

Resultant,  7. 

Wheel  and  axle,  30. 

manometer,  172. 

Rigid  body,  5. 

Wheel-barometer,  152. 

Natural  history  and  natural  phi- 

Rotating vessel  of  liquid,  93. 

Whirling  vessel  of  liquid,  95. 
Work,  22-25. 

losophy,  i. 
Needles  floating,  103. 

Screw,  and  screw-press,  33. 
Second  law  of  motion,  42. 

—  in  producing  motion,  52. 
—  ,  principle  of,  25. 

Newton's  experiments  with  pen- 

Sensibility and  instability,  38. 

Wrench,  15. 

dulums,  50. 
—  laws  of  motion,  41-45. 
Nicholson's  hydrometer,  in. 

Sensibility  of  balance,  35. 
Simple-harmonic  motion,  65. 
Simple  pendulum,  62. 

Young's  modulus,  78. 
Zero,  errors  of,  151. 

REGIONAL  LIBRARY  FA 


000  940  772     7 


-'"CALIFORNIA, 

LIBRARY 


